distrib

cdf_bernoulli (x, p) — Function

Returns the value at x of the cumulative distribution function of a ${\it Bernoulli}(p)$ random variable, with $0 \leq p \leq 1$. To make use of this function, write first load("distrib").

The cdf is

$$F(x; p) = I_{1-p}(1-\lfloor x \rfloor, \lfloor x \rfloor + 1)$$

$$F(x; p) = I_{1-p}(1-\lfloor x \rfloor, \lfloor x \rfloor + 1)$$

cdf_beta (x, a, b) — Function

Returns the value at x of the cumulative distribution function of a ${\it Beta}(a,b)$ random variable, with $a,b>0$.

The cdf is

$$F(x; a, b) = \cases{ 0 & $x < 0$ \cr I_x(a,b) & $0 \le x \le 1$ \cr 1 & $x > 1$ }$$

$$F(x; a, b) = \cases{ 0 & $x < 0$ \cr I_x(a,b) & $0 \le x \le 1$ \cr 1 & $x > 1$ }$$

(%i1) load ("distrib")$

(%i2) cdf_beta(1/3,15,2);
                               11
(%o2)                       --------
                            14348907


(%i3) float(%);
(%o3)                 7.666089131388195e-7

cdf_binomial (x, n, p) — Function

Returns the value at x of the cumulative distribution function of a ${\it Binomial}(n,p)$ random variable, with $0 \leq p \leq 1$ and $n$ a positive integer.

The cdf is

$$F(x; n, p) = I_{1-p}(n-\lfloor x \rfloor, \lfloor x \rfloor + 1)$$

$$F(x; n, p) = I_{1-p}(n-\lfloor x \rfloor, \lfloor x \rfloor + 1)$$

where $I_z(a,b)$ is the beta_005fincomplete_005fregularized function.

(%i1) load ("distrib")$

(%i2) cdf_binomial(5,7,1/6);
                              7775
(%o2)                         ----
                              7776


(%i3) float(%);
(%o3)                  0.9998713991769548

See also: beta_incomplete_regularized.

cdf_cauchy (x, a, b) — Function

Returns the value at x of the cumulative distribution function of a ${\it Cauchy}(a,b)$ random variable, with $b>0$. To make use of this function, write first load("distrib").

The cdf is

$$F(x; a, b) = {1\over 2} + {1\over \pi} \tan^{-1} {x-a\over b}$$

$$F(x; a, b) = {1\over 2} + {1\over \pi} \tan^{-1} {x-a\over b}$$

cdf_chi2 (x, n) — Function

Returns the value at $x$ of the cumulative distribution function of a Chi-square random variable $\chi^2(n)$ , with $n>0$.

The cdf is

$$F(x; n) = \cases{ 1 - Q\left(\displaystyle{n\over 2}, {x\over 2}\right) & $x > 0$ \cr 0 & otherwise }$$

$$F(x; n) = \cases{ 1 - Q\left(\displaystyle{n\over 2}, {x\over 2}\right) & $x > 0$ \cr 0 & otherwise }$$

where $Q(a,z)$ is the gamma_005fincomplete_005fregularized function.

(%i1) load ("distrib")$

(%i2) cdf_chi2(3,4);
                                                 3
(%o2)        1 - gamma_incomplete_regularized(2, -)
                                                 2


(%i3) float(%);
(%o3)                  0.44217459962892525

See also: gamma_incomplete_regularized.

cdf_continuous_uniform (x, a, b) — Function

Returns the value at x of the cumulative distribution function of a ${\it ContinuousUniform}(a,b)$ random variable, with $a \lt b.$ To make use of this function, write first load("distrib").

The cdf is

$$F(x; a, b) = \cases{ 0 & for $x < a$ \cr \cr \displaystyle{x-a\over b-a} & for $a \le x \le b$ \cr \cr 1 & for $x > b$ }$$

$$F(x; a, b) = \cases{ 0 & for $x < a$ \cr \cr \displaystyle{x-a\over b-a} & for $a \le x \le b$ \cr \cr 1 & for $x > b$ }$$

cdf_discrete_uniform (x, n) — Function

Returns the value at x of the cumulative distribution function of a ${\it DiscreteUniform}(n)$ random variable, with $n$ a strictly positive integer. To make use of this function, write first load("distrib").

The cdf is

$$F(x; n) = {\lfloor x \rfloor \over n}$$

$$F(x; n) = {\lfloor x \rfloor \over n}$$

cdf_exp (x, m) — Function

Returns the value at x of the cumulative distribution function of an ${\it Exponential}(m)$ random variable, with $m>0$.

The ${\it Exponential}(m)$ random variable is equivalent to the ${\it Weibull}(1,1/m)$ .

The cdf is

$$F(x; m) = \cases{ 1 - e^{-mx} & $x \ge 0$ \cr 0 & otherwise }$$

$$F(x; m) = \cases{ 1 - e^{-mx} & $x \ge 0$ \cr 0 & otherwise }$$

(%i1) load ("distrib")$

(%i2) cdf_exp(x,m);
                          - m x
(%o2)              (1 - %e     ) unit_step(x)

cdf_f (x, m, n) — Function

Returns the value at x of the cumulative distribution function of a F random variable $F(m,n)$, with $m,n>0$.

The cdf is

$$F(x; m, n) = \cases{ 1 - I_z\left(\displaystyle{m\over 2}, {n\over 2}\right) & $x > 0$ \cr 0 & otherwise }$$

$$F(x; m, n) = \cases{ 1 - I_z\left(\displaystyle{m\over 2}, {n\over 2}\right) & $x > 0$ \cr 0 & otherwise }$$

where

$$z = {n\over mx+n}$$

$$z = {n\over mx+n}$$

and $I_z(a,b)$ is the beta_005fincomplete_005fregularized function.

(%i1) load ("distrib")$

(%i2) cdf_f(2,3,9/4);
                                            9  3  3
(%o2)       1 - beta_incomplete_regularized(-, -, --)
                                            8  2  11


(%i3) float(%);
(%o3)                  0.6675672817900802

See also: beta_incomplete_regularized.

cdf_gamma (x, a, b) — Function

Returns the value at x of the cumulative distribution function of a $\Gamma\left(a,b\right)$ random variable, with $a,b>0$.

The cdf is

$$F(x; a, b) = \cases{ 1-Q(a,{x\over b}) & for $x \ge 0$ \cr \cr 0 & for $x < 0$ }$$

$$F(x; a, b) = \cases{ 1-Q(a,{x\over b}) & for $x \ge 0$ \cr \cr 0 & for $x < 0$ }$$

where $Q(a,z)$ is the gamma_005fincomplete_005fregularized function.

(%i1) load ("distrib")$

(%i2) cdf_gamma(3,5,21);
                                                 1
(%o2)        1 - gamma_incomplete_regularized(5, -)
                                                 7


(%i3) float(%);
(%o3)                 4.402663157376807e-7

See also: gamma_incomplete_regularized.

cdf_general_finite_discrete (x, v) — Function

Returns the value at x of the cumulative distribution function of a general finite discrete random variable, with vector probabilities $v$.

See pdf_005fgeneral_005ffinite_005fdiscrete for more details.

(%i1) load ("distrib")$

(%i2) cdf_general_finite_discrete(2, [1/7, 4/7, 2/7]);
                                5
(%o2)                           -
                                7


(%i3) cdf_general_finite_discrete(2, [1, 4, 2]);
                                5
(%o3)                           -
                                7


(%i4) cdf_general_finite_discrete(2+1/2, [1, 4, 2]);
                                5
(%o4)                           -
                                7

See also: pdf_general_finite_discrete.

cdf_geometric (x, p) — Function

Returns the value at x of the cumulative distribution function of a ${\it Geometric}(p)$ random variable, with $0 < p \leq 1$

The cdf is

$$1-(1-p)^{1 + \lfloor x \rfloor}$$

$$1-(1-p)^{1 + \lfloor x \rfloor}$$

load("distrib") loads this function.

cdf_gumbel (x, a, b) — Function

Returns the value at x of the cumulative distribution function of a ${\it Gumbel}(a,b)$ random variable, with $b>0$. To make use of this function, write first load("distrib").

The cdf is

$$F(x; a, b) = \exp\left[-\exp\left({a-x\over b}\right)\right]$$

$$F(x; a, b) = \exp\left[-\exp\left({a-x\over b}\right)\right]$$

cdf_hypergeometric (x, n_1, n_2, n) — Function

Returns the value at x of the cumulative distribution function of a ${\it Hypergeometric}(n1,n2,n)$ random variable, with $n_1$, $n_2$ and $n$ non negative integers and $n\leq n_1+n_2$. See pdf_hypergeometric for a more complete description.

To make use of this function, write first load("distrib").

The cdf is

$$F(x; n_1, n_2, n) = {n_2+n_1\choose n}^{-1} \sum_{k=0}^{\lfloor x \rfloor} {n_1 \choose k} {n_2 \choose n - k}$$

$$F(x; n_1, n_2, n) = {n_2+n_1\choose n}^{-1} \sum_{k=0}^{\lfloor x \rfloor} {n_1 \choose k} {n_2 \choose n - k}$$

cdf_laplace (x, a, b) — Function

Returns the value at x of the cumulative distribution function of a ${\it Laplace}(a,b)$ random variable, with $b>0$. To make use of this function, write first load("distrib").

The cdf is

$$F(x; a, b) = \cases{ \displaystyle{1\over 2} \exp\left({x-a\over b}\right) & for $x < a$\cr \cr 1-\displaystyle{1\over 2} \exp\left({x-a\over b}\right) & for $x \ge a$ }$$

$$F(x; a, b) = \cases{ \displaystyle{1\over 2} \exp\left({x-a\over b}\right) & for $x < a$\cr \cr 1-\displaystyle{1\over 2} \exp\left({x-a\over b}\right) & for $x \ge a$ }$$

cdf_logistic (x, a, b) — Function

Returns the value at x of the cumulative distribution function of a ${\it Logistic}(a,b)$ random variable , with $b>0$. To make use of this function, write first load("distrib").

The cdf is

$$F(x; a, b) = {1\over 1+e^{-(x-a)/b}}$$

$$F(x; a, b) = {1\over 1+e^{-(x-a)/b}}$$

cdf_lognormal (x, m, s) — Function

Returns the value at x of the cumulative distribution function of a ${\it Lognormal}(m,s)$ random variable, with $s>0$. This function is defined in terms of Maxima’s built-in error function erf.

The cdf is

$$F(x; m, s) = \cases{ \displaystyle{1\over 2}\left[1+{\rm erf}\left({\log x - m\over s\sqrt{2}}\right)\right] & for $x > 0$ \cr \cr 0 & for $x \le 0$ }$$

$$F(x; m, s) = \cases{ \displaystyle{1\over 2}\left[1+{\rm erf}\left({\log x - m\over s\sqrt{2}}\right)\right] & for $x > 0$ \cr \cr 0 & for $x \le 0$ }$$

(%i1) load ("distrib")$

(%i2) cdf_lognormal(x,m,s);
                                 log(x) - m
                             erf(----------)
                                 sqrt(2) s     1
(%o2)          unit_step(x) (--------------- + -)
                                    2          2

See also erf.

See also: erf.

cdf_negative_binomial (x, n, p) — Function

Returns the value at x of the cumulative distribution function of a ${\it NegativeBinomial}(n,p)$ random variable, with $0 < p \leq 1$ and $n$ a positive number.

The cdf is

$$F(x; n, p) = I_p(n,\lfloor x \rfloor + 1)$$

$$F(x; n, p) = I_p(n,\lfloor x \rfloor + 1)$$

where $I_p(a,b)$ is the beta_005fincomplete_005fregularized function.

(%i1) load ("distrib")$

(%i2) cdf_negative_binomial(3,4,1/8);
                              3271
(%o2)                        ------
                             524288

See also: beta_incomplete_regularized.

cdf_noncentral_chi2 (x, n, ncp) — Function

Returns the value at x of the cumulative distribution function of a noncentral Chi-square random variable m4_noncentral_chi2(n,ncp) , with $n>0$ and noncentrality parameter $ncp \ge 0.$ To make use of this function, write first load("distrib").

cdf_noncentral_student_t (x, n, ncp) — Function

Returns the value at x of the cumulative distribution function of a noncentral Student random variable ${\it nc_t}(n, ncp)$ , with $n>0$ degrees of freedom and noncentrality parameter $ncp$. This function has no closed form and it is numerically computed.

(%i1) load ("distrib")$

(%i2) cdf_noncentral_student_t(-2,5,-5);
(%o2)                   0.995203009331975

cdf_normal (x, m, s) — Function

Returns the value at x of the cumulative distribution function of a ${\it Normal}(m, s)$ random variable, with $s>0$. This function is defined in terms of Maxima’s built-in error function erf.

The cdf can be written analytically:

$$F(x; m, s) = {1\over 2} + {1\over 2} {\rm erf}\left(x-m\over s\sqrt{2}\right)$$

$$F(x; m, s) = {1\over 2} + {1\over 2} {\rm erf}\left(x-m\over s\sqrt{2}\right)$$

(%i1) load ("distrib")$

(%i2) cdf_normal(x,m,s);
                             x - m
                       erf(---------)
                           sqrt(2) s    1
(%o2)                  -------------- + -
                             2          2

See also erf.

See also: erf.

cdf_pareto (x, a, b) — Function

Returns the value at x of the cumulative distribution function of a ${\it Pareto}(a,b)$ random variable, with $a,b>0$. To make use of this function, write first load("distrib").

The cdf is

$$F(x; a, b) = \cases{ 1-\left(\displaystyle{b\over x}\right)^a & for $x \ge b$\cr 0 & for $x < b$ }$$

$$F(x; a, b) = \cases{ 1-\left(\displaystyle{b\over x}\right)^a & for $x \ge b$\cr 0 & for $x < b$ }$$

cdf_poisson (x, m) — Function

Returns the value at x of the cumulative distribution function of a ${\it Poisson}(m)$ random variable, with $m>0$.

The cdf is

$$F(x; m) = Q(\lfloor x \rfloor + 1, m)$$

$$F(x; m) = Q(\lfloor x \rfloor + 1, m)$$

where $Q(x,m)$ is the gamma_005fincomplete_005fregularized function.

(%i1) load ("distrib")$

(%i2) cdf_poisson(3,5);
(%o2)          gamma_incomplete_regularized(4, 5)


(%i3) float(%);
(%o3)                  0.26502591529736186

See also: gamma_incomplete_regularized.

cdf_rayleigh (x, b) — Function

Returns the value at x of the cumulative distribution function of a ${\it Rayleigh}(b)$ random variable, with $b>0$.

The ${\it Rayleigh}(b)$ random variable is equivalent to the ${\it Weibull}(2,1/b)$ .

The cdf is

$$F(x; b) = \cases{ 1 - e^{-b^2 x^2} & for $x \ge 0$\cr 0 & for $x < 0$ }$$

$$F(x; b) = \cases{ 1 - e^{-b^2 x^2} & for $x \ge 0$\cr 0 & for $x < 0$ }$$

(%i1) load ("distrib")$

(%i2) cdf_rayleigh(x,b);
                            2  2
                         - b  x
(%o2)             (1 - %e       ) unit_step(x)

cdf_student_t (x, n) — Function

Returns the value at x of the cumulative distribution function of a Student random variable $t(n)$ , with $n>0$ degrees of freedom.

The cdf is

$$F(x; n) = \cases{ 1-\displaystyle{1\over 2} I_t\left({n\over 2}, {1\over 2}\right) & $x \ge 0$ \cr \cr \displaystyle{1\over 2} I_t\left({n\over 2}, {1\over 2}\right) & $x < 0$ }$$

$$F(x; n) = \cases{ 1-\displaystyle{1\over 2} I_t\left({n\over 2}, {1\over 2}\right) & $x \ge 0$ \cr \cr \displaystyle{1\over 2} I_t\left({n\over 2}, {1\over 2}\right) & $x < 0$ }$$

where $t = n/(n+x^2)$ and $I_t(a,b)$ is the beta_005fincomplete_005fregularized function.

(%i1) load ("distrib")$

(%i2) cdf_student_t(1/2, 7/3);
                                            7  1  28
                beta_incomplete_regularized(-, -, --)
                                            6  2  31
(%o2)       1 - -------------------------------------
                                  2


(%i3) float(%);
(%o3)                  0.6698450596140415

See also: beta_incomplete_regularized.

cdf_weibull (x, a, b) — Function

Returns the value at x of the cumulative distribution function of a ${\it Weibull}(a,b)$ random variable, with $a,b>0$. To make use of this function, write first load("distrib").

The cdf is

$$F(x; a, b) = \cases{ 1 - e^{-(x/b)^a} & for $x \ge 0$ \cr 0 & for $x < 0$ }$$

$$F(x; a, b) = \cases{ 1 - e^{-(x/b)^a} & for $x \ge 0$ \cr 0 & for $x < 0$ }$$

kurtosis_bernoulli (p) — Function

Returns the kurtosis coefficient of a ${\it Bernoulli}(p)$ random variable, with $0 \leq p \leq 1$.

The ${\it Bernoulli}(p)$ random variable is equivalent to the ${\it Binomial}(1,p)$ .

The kurtosis coefficient is

$$KU[X] = {1-6p(1-p) \over p(1-p)}$$

$$KU[X] = {1-6p(1-p) \over p(1-p)}$$

(%i1) load ("distrib")$

(%i2) kurtosis_bernoulli(p);
                         1 - 6 (1 - p) p
(%o2)                    ---------------
                            (1 - p) p

kurtosis_beta (a, b) — Function

Returns the kurtosis coefficient of a ${\it Beta}(a,b)$ random variable, with $a,b>0$. To make use of this function, write first load("distrib").

The kurtosis coefficient is

$$KU[X] = {3(a+b+1)\left(2(a+b)^2+ab(a+b-6)\right) \over ab(a+b+2)(a+b+3)} - 3$$

$$KU[X] = {3(a+b+1)\left(2(a+b)^2+ab(a+b-6)\right) \over ab(a+b+2)(a+b+3)} - 3$$

kurtosis_binomial (n, p) — Function

Returns the kurtosis coefficient of a ${\it Binomial}(n,p)$ random variable, with $0 \leq p \leq 1$ and $n$ a positive integer. To make use of this function, write first load("distrib").

The kurtosis coefficient is

$$KU[X] = {1-6p(1-p)\over np(1-p)}$$

$$KU[X] = {1-6p(1-p)\over np(1-p)}$$

kurtosis_chi2 (n) — Function

Returns the kurtosis coefficient of a Chi-square random variable $\chi^2(n)$ , with $n>0$.

The $\chi^2(n)$ random variable is equivalent to the $\Gamma\left(n/2,2\right)$ .

The kurtosis coefficient is

$$KU[X] = {12\over n}$$

$$KU[X] = {12\over n}$$

(%i1) load ("distrib")$

(%i2) kurtosis_chi2(n);
                               12
(%o2)                          --
                               n

kurtosis_continuous_uniform (a, b) — Function

Returns the kurtosis coefficient of a ${\it ContinuousUniform}(a,b)$ random variable, with $a \lt b.$ To make use of this function, write first load("distrib").

The kurtosis coefficient is

$$KU[X] = -{6\over5}$$

$$KU[X] = -{6\over5}$$

kurtosis_discrete_uniform (n) — Function

Returns the kurtosis coefficient of a ${\it DiscreteUniform}(n)$ random variable, with $n$ a strictly positive integer. To make use of this function, write first load("distrib").

The kurtosis coefficient is

$$KU[X] = - {6(n^2+1)\over 5 (n^2-1)}$$

$$KU[X] = - {6(n^2+1)\over 5 (n^2-1)}$$

kurtosis_exp (m) — Function

Returns the kurtosis coefficient of an ${\it Exponential}(m)$ random variable, with $m>0$.

The ${\it Exponential}(m)$ random variable is equivalent to the ${\it Weibull}(1,1/m)$ .

The kurtosis coefficient is

$$KU[X] = 6$$

$$KU[X] = 6$$

(%i1) load ("distrib")$

(%i2) kurtosis_exp(m);
(%o2)                           6

kurtosis_f (m, n) — Function

Returns the kurtosis coefficient of a F random variable $F(m,n)$, with $m>0, n>8$. To make use of this function, write first load("distrib").

The kurtosis coefficient is

$$KU[X] = 12{m(n+m-2)(5n-22) + (n-4)(n-2)^2 \over m(n-8)(n-6)(n+m-2)}$$

$$KU[X] = 12{m(n+m-2)(5n-22) + (n-4)(n-2)^2 \over m(n-8)(n-6)(n+m-2)}$$

kurtosis_gamma (a, b) — Function

Returns the kurtosis coefficient of a $\Gamma\left(a,b\right)$ random variable, with $a,b>0$. To make use of this function, write first load("distrib").

The kurtosis coefficient is

$$KU[X] = {6\over a}$$

$$KU[X] = {6\over a}$$

kurtosis_general_finite_discrete (v) — Function

Returns the kurtosis coefficient of a general finite discrete random variable, with vector probabilities $v$.

See pdf_005fgeneral_005ffinite_005fdiscrete for more details.

See also: pdf_general_finite_discrete.

kurtosis_geometric (p) — Function

Returns the kurtosis coefficient of a geometric random variable
${\it Geometric}(p)$ , with $0 < p \leq 1$.

The kurtosis coefficient is

$$KU[X] = {p^2-6p+6 \over 1-p}$$

$$KU[X] = {p^2-6p+6 \over 1-p}$$

load("distrib") loads this function.

kurtosis_gumbel (a, b) — Function

Returns the kurtosis coefficient of a ${\it Gumbel}(a,b)$ random variable, with $b>0$. To make use of this function, write first load("distrib").

The kurtosis coefficient is

$$KU[X] = {12\over 5}$$

$$KU[X] = {12\over 5}$$

kurtosis_hypergeometric (n_1, n_2, n) — Function

Returns the kurtosis coefficient of a ${\it Hypergeometric}(n_1,n_2,n)$ random variable, with $n_1$, $n_2$ and $n$ non negative integers and $n\leq n1+n2$. To make use of this function, write first load("distrib").

The kurtosis coefficient is

$$\eqalign{ KU[X] = & \left[{C(1)C(0)^2 \over n n_1 n_2 C(3)C(2)C(n)}\right. \cr & \times \left.\left( {3n_1n_2\left((n-2)C(0)^2+6nC(n)-n^2C(0)\right) \over C(0)^2 } -6nC(n) + C(0)C(-1) \right)\right] \cr &-3 }$$

$$ \eqalign{ KU[X] = & \left[{C(1)C(0)^2 \over n n_1 n_2 C(3)C(2)C(n)}\right. \cr & \times \left.\left( {3n_1n_2\left((n-2)C(0)^2+6nC(n)-n^2C(0)\right) \over C(0)^2 } -6nC(n) + C(0)C(-1) \right)\right] \cr &-3 }$$

where $C(k) = n_1+n_2-k.$

kurtosis_laplace (a, b) — Function

Returns the kurtosis coefficient of a ${\it Laplace}(a,b)$ random variable, with $b>0$. To make use of this function, write first load("distrib").

The kurtosis coefficient is

$$KU[X] = 3$$

$$KU[X] = 3$$

kurtosis_logistic (a, b) — Function

Returns the kurtosis coefficient of a ${\it Logistic}(a,b)$ random variable, with $b>0$. To make use of this function, write first load("distrib").

The kurtosis coefficient is

$$KU[X] = {6\over 5}$$

$$KU[X] = {6\over 5}$$

kurtosis_lognormal (m, s) — Function

Returns the kurtosis coefficient of a ${\it Lognormal}(m,s)$ random variable, with $s>0$. To make use of this function, write first load("distrib").

The kurtosis coefficient is

$$KU[X] = \exp\left(4s^2\right)+2\exp\left(3s^2\right)+3\exp\left(2s^2\right)-3$$

$$KU[X] = \exp\left(4s^2\right)+2\exp\left(3s^2\right)+3\exp\left(2s^2\right)-3$$

kurtosis_negative_binomial (n, p) — Function

Returns the kurtosis coefficient of a ${\it NegativeBinomial}(n,p)$ random variable, with $0 < p \leq 1$ and $n$ a positive number. To make use of this function, write first load("distrib").

The kurtosis coefficient is

$$KU[X] = {p^2-6p+6 \over n(1-p)}$$

$$KU[X] = {p^2-6p+6 \over n(1-p)}$$

kurtosis_noncentral_chi2 (n, ncp) — Function

Returns the kurtosis coefficient of a noncentral Chi-square random variable m4_noncentral_chi2(n,ncp) , with $n>0$ and noncentrality parameter $ncp \ge 0.$

The kurtosis coefficient is

$$KU[X] = {12(n+4\mu)\over (2+2\mu)^2}$$

$$KU[X] = {12(n+4\mu)\over (2+2\mu)^2}$$

where $\mu$ is the noncentrality parameter ncp.

kurtosis_noncentral_student_t (n, ncp) — Function

Returns the kurtosis coefficient of a noncentral Student random variable ${\it nc_t}(n, ncp)$ , with $n>4$ degrees of freedom and noncentrality parameter $ncp$. To make use of this function, write first load("distrib").

If $U$ is a non-central Student’s $t$ random variable with $n$ degrees of freedom and a noncentrality parameter $\mu,$ the kurtosis is

$$\eqalign{ KU[U] &= {\mu_4\over \sigma^4} - 3\cr \mu_4 &= {{\left(\mu^4+6\mu^2+3\right)n^2}\over{(n-4)(n-2)}} -\left({{n\left(3(3n-5)+\mu^2(n+1)\right) }\over{(n-3)(n-2)}}-3\sigma^2\right) F \cr \sigma^2 &= {{n\left(\mu^2+1\right)}\over{n-2}}-{{n \mu^2 \Gamma\left({{n-1}\over{2}}\right)^2}\over{2\Gamma\left({{n }\over{2}}\right)^2}} \cr F &= {n\mu^2\Gamma\left({n-1\over 2}\right)^2 \over 2\sigma^4\Gamma\left({n\over 2}\right)^2} }$$

$$\eqalign{ KU[U] &= {\mu_4\over \sigma^4} - 3\cr \mu_4 &= {{\left(\mu^4+6\mu^2+3\right)n^2}\over{(n-4)(n-2)}} -\left({{n\left(3(3n-5)+\mu^2(n+1)\right) }\over{(n-3)(n-2)}}-3\sigma^2\right) F \cr \sigma^2 &= {{n\left(\mu^2+1\right)}\over{n-2}}-{{n \mu^2 \Gamma\left({{n-1}\over{2}}\right)^2}\over{2\Gamma\left({{n }\over{2}}\right)^2}} \cr F &= {n\mu^2\Gamma\left({n-1\over 2}\right)^2 \over 2\sigma^4\Gamma\left({n\over 2}\right)^2} }$$

kurtosis_normal (m, s) — Function

Returns the kurtosis coefficient of a ${\it Normal}(m, s)$ random variable, with $s>0$, which is always equal to 0. To make use of this function, write first load("distrib").

The kurtosis coefficient is

$$KU[X] = 0$$

$$KU[X] = 0$$

kurtosis_pareto (a, b) — Function

Returns the kurtosis coefficient of a ${\it Pareto}(a,b)$ random variable, with $a>4,b>0$. To make use of this function, write first load("distrib").

The kurtosis coefficient is

$$KU[X] = {6\left(a^3+a^2-6*a-2\right) \over a(a-3)(a-4)} - 3$$

$$KU[X] = {6\left(a^3+a^2-6*a-2\right) \over a(a-3)(a-4)} - 3$$

kurtosis_poisson (m) — Function

Returns the kurtosis coefficient of a Poisson random variable $Poi(m)$, with $m>0$. To make use of this function, write first load("distrib").

The kurtosis coefficient is

$$KU[X] = {1\over m}$$

$$KU[X] = {1\over m}$$

kurtosis_rayleigh (b) — Function

Returns the kurtosis coefficient of a ${\it Rayleigh}(b)$ random variable, with $b>0$.

The ${\it Rayleigh}(b)$ random variable is equivalent to the ${\it Weibull}(2,1/b)$ .

The kurtosis coefficient is

$$KU[X] = {32-3\pi\over (4-\pi)^2} - 3$$

$$KU[X] = {32-3\pi\over (4-\pi)^2} - 3$$

(%i1) load ("distrib")$

(%i2) kurtosis_rayleigh(b);
                                  2
                             3 %pi
                         2 - ------
                               16
(%o2)                    ---------- - 3
                              %pi 2
                         (1 - ---)
                               4

kurtosis_student_t (n) — Function

Returns the kurtosis coefficient of a Student random variable $t(n)$ , with $n>4$. To make use of this function, write first load("distrib").

The kurtosis coefficient is

$$KU[X] = {6\over n-4}$$

$$KU[X] = {6\over n-4}$$

kurtosis_weibull (a, b) — Function

Returns the kurtosis coefficient of a ${\it Weibull}(a,b)$ random variable, with $a,b>0$. To make use of this function, write first load("distrib").

The kurtosis coefficient is

$$KU[X] = { \Gamma_4 - 4\Gamma_1 \Gamma_3 + 6\Gamma_1^2 \Gamma_2 - 3 \Gamma_1^4 \over \left[\Gamma_2 - \Gamma_1^2\right]^2 } - 3$$

$$KU[X] = { \Gamma_4

• 4\Gamma_1 \Gamma_3

• 6\Gamma_1^2 \Gamma_2

• 3 \Gamma_1^4 \over \left[\Gamma_2 - \Gamma_1^2\right]^2 } - 3$$

where $\Gamma_k = \Gamma\left(1+k/a\right).$

mean_bernoulli (p) — Function

Returns the mean of a ${\it Bernoulli}(p)$ random variable, with $0 \leq p \leq 1$.

The ${\it Bernoulli}(p)$ random variable is equivalent to the ${\it Binomial}(1,p)$ .

The mean is

$$E[X] = p$$

$$E[X] = p$$

(%i1) load ("distrib")$

(%i2) mean_bernoulli(p);
(%o2)                           p

mean_beta (a, b) — Function

Returns the mean of a ${\it Beta}(a,b)$ random variable, with $a,b>0$. To make use of this function, write first load("distrib").

The mean is

$$E[X] = {a\over a+b}$$

$$E[X] = {a\over a+b}$$

mean_binomial (n, p) — Function

Returns the mean of a ${\it Binomial}(n,p)$ random variable, with $0 \leq p \leq 1$ and $n$ a positive integer. To make use of this function, write first load("distrib").

The mean is

$$E[X] = np$$

$$E[X] = np$$

mean_chi2 (n) — Function

Returns the mean of a Chi-square random variable $\chi^2(n)$ , with $n>0$.

The $\chi^2(n)$ random variable is equivalent to the $\Gamma\left(n/2,2\right)$ .

The mean is

$$E[X] = n$$

$$E[X] = n$$

(%i1) load ("distrib")$

(%i2) mean_chi2(n);
(%o2)                           n

mean_continuous_uniform (a, b) — Function

Returns the mean of a ${\it ContinuousUniform}(a,b)$ random variable, with $a \lt b.$ To make use of this function, write first load("distrib").

The mean is

$$E[X] = {a+b\over 2}$$

$$E[X] = {a+b\over 2}$$

mean_discrete_uniform (n) — Function

Returns the mean of a ${\it DiscreteUniform}(n)$ random variable, with $n$ a strictly positive integer. To make use of this function, write first load("distrib").

The mean is

$$E[X] = {n+1\over 2}$$

$$E[X] = {n+1\over 2}$$

mean_exp (m) — Function

Returns the mean of an ${\it Exponential}(m)$ random variable, with $m>0$.

The ${\it Exponential}(m)$ random variable is equivalent to the ${\it Weibull}(1,1/m)$ .

The mean is

$$E[X] = {1\over m}$$

$$E[X] = {1\over m}$$

(%i1) load ("distrib")$

(%i2) mean_exp(m);
                                1
(%o2)                           -
                                m

mean_f (m, n) — Function

Returns the mean of a F random variable $F(m,n)$, with $m>0, n>2$. To make use of this function, write first load("distrib").

The mean is

$$E[X] = {n\over n-2}$$

$$E[X] = {n\over n-2}$$

mean_gamma (a, b) — Function

Returns the mean of a $\Gamma\left(a,b\right)$ random variable, with $a,b>0$. To make use of this function, write first load("distrib").

The mean is

$$E[X] = ab$$

$$E[X] = ab$$

mean_general_finite_discrete (v) — Function

Returns the mean of a general finite discrete random variable, with vector probabilities $v$.

See pdf_005fgeneral_005ffinite_005fdiscrete for more details.

See also: pdf_general_finite_discrete.

mean_geometric (p) — Function

Returns the mean of a ${\it Geometric}(p)$ random variable, with $0 < p \leq 1$.

The mean is

$$E[X] = {1\over p} - 1$$

$$E[X] = {1\over p} - 1$$

The probability from which the mean is derived is defined as $p (1 - p)^x$. This is interpreted as the probability of $x$ failures before the first success.

load("distrib") loads this function.

mean_gumbel (a, b) — Function

Returns the mean of a ${\it Gumbel}(a,b)$ random variable, with $b>0$.

The mean is

$$E[X] = a+b\gamma$$

$$E[X] = a+b\gamma$$

(%i1) load ("distrib")$

(%i2) mean_gumbel(a,b);
(%o2)                     %gamma b + a

where symbol %gamma stands for the Euler-Mascheroni constant. See also _0025gamma.

See also: %gamma.

mean_hypergeometric (n_1, n_2, n) — Function

Returns the mean of a discrete uniform random variable ${\it Hypergeometric}(n_1,n_2,n)$ , with $n_1$, $n_2$ and $n$ non negative integers and $n\leq n_1+n_2$. To make use of this function, write first load("distrib").

The mean is

$$E[X] = {n n_1\over n_2+n_1}$$

$$E[X] = {n n_1\over n_2+n_1}$$

mean_laplace (a, b) — Function

Returns the mean of a ${\it Laplace}(a,b)$ random variable, with $b>0$. To make use of this function, write first load("distrib").

The mean is

$$E[X] = a$$

$$E[X] = a$$

mean_logistic (a, b) — Function

Returns the mean of a ${\it Logistic}(a,b)$ random variable , with $b>0$. To make use of this function, write first load("distrib").

The mean is

$$E[X] = a$$

$$E[X] = a$$

mean_lognormal (m, s) — Function

Returns the mean of a ${\it Lognormal}(m,s)$ random variable, with $s>0$. To make use of this function, write first load("distrib").

The mean is

$$E[X] = \exp\left(m+{s^2\over 2}\right)$$

$$E[X] = \exp\left(m+{s^2\over 2}\right)$$

mean_negative_binomial (n, p) — Function

Returns the mean of a ${\it NegativeBinomial}(n,p)$ random variable, with $0 < p \leq 1$ and $n$ a positive number. To make use of this function, write first load("distrib").

The mean is

$$E[X] = {n(1-p)\over p}$$

$$E[X] = {n(1-p)\over p}$$

mean_noncentral_chi2 (n, ncp) — Function

Returns the mean of a noncentral Chi-square random variable m4_noncentral_chi2(n,ncp) , with $n>0$ and noncentrality parameter $ncp \ge 0.$

The mean is

$$E[X] = n + \mu$$

$$E[X] = n + \mu$$

where $\mu$ is the noncentrality parameter ncp.

mean_noncentral_student_t (n, ncp) — Function

Returns the mean of a noncentral Student random variable ${\it nc_t}(n, ncp)$ , with $n>1$ degrees of freedom and noncentrality parameter $ncp$. To make use of this function, write first load("distrib").

The mean is

$$E[X] = {\mu \sqrt{n}; \Gamma\left(\displaystyle{n-1\over 2}\right) \over \sqrt{2};\Gamma\left(\displaystyle{n\over 2}\right)}$$

$$E[X] = {\mu \sqrt{n}; \Gamma\left(\displaystyle{n-1\over 2}\right) \over \sqrt{2};\Gamma\left(\displaystyle{n\over 2}\right)}$$

where $\mu$ is the noncentrality parameter $ncp$.

(%i1) load ("distrib")$

(%i2) mean_noncentral_student_t(df,k);
                          df - 1
                    gamma(------) sqrt(df) k
                            2
(%o2)               ------------------------
                                     df
                       sqrt(2) gamma(--)
                                     2

mean_normal (m, s) — Function

Returns the mean of a ${\it Normal}(m, s)$ random variable, with $s>0$. To make use of this function, write first load("distrib").

The mean is

$$E[X] = m$$

$$E[X] = m$$

mean_pareto (a, b) — Function

Returns the mean of a ${\it Pareto}(a,b)$ random variable, with $a>1,b>0$. To make use of this function, write first load("distrib").

The mean is

$$E[X] = {ab\over a-1}$$

$$E[X] = {ab\over a-1}$$

mean_poisson (m) — Function

Returns the mean of a ${\it Poisson}(m)$ random variable, with $m>0$. To make use of this function, write first load("distrib").

The mean is

$$E[X] = m$$

$$E[X] = m$$

mean_rayleigh (b) — Function

Returns the mean of a ${\it Rayleigh}(b)$ random variable, with $b>0$.

The ${\it Rayleigh}(b)$ random variable is equivalent to the ${\it Weibull}(2,1/b)$ .

The mean is

$$E[X] = {\sqrt{\pi}\over 2b}$$

$$E[X] = {\sqrt{\pi}\over 2b}$$

(%i1) load ("distrib")$

(%i2) mean_rayleigh(b);
                            sqrt(%pi)
(%o2)                       ---------
                               2 b

mean_student_t (n) — Function

Returns the mean of a Student random variable $t(n)$ , with $n>0$. To make use of this function, write first load("distrib").

The mean is

$$E[X] = 0$$

$$E[X] = 0$$

mean_weibull (a, b) — Function

Returns the mean of a ${\it Weibull}(a,b)$ random variable, with $a,b>0$. To make use of this function, write first load("distrib").

The mean is

$$E[X] = b\Gamma\left(1+{1\over a}\right)$$

$$E[X] = b\Gamma\left(1+{1\over a}\right)$$

pdf_bernoulli (x, p) — Function

Returns the value at x of the probability function of a ${\it Bernoulli}(p)$ random variable, with $0 \leq p \leq 1$.

The ${\it Bernoulli}(p)$ random variable is equivalent to the ${\it Binomial}(1,p)$ .

The mean is

$$f(x; p) = p^x (1-p)^{1-x}$$

$$f(x; p) = p^x (1-p)^{1-x}$$

(%i1) load ("distrib")$

(%i2) pdf_bernoulli(1,p);
(%o2) if equal(p, 0) then 0 elseif equal(p, 1) then 1 else p

pdf_beta (x, a, b) — Function

Returns the value at x of the density function of a ${\it Beta}(a,b)$ random variable, with $a,b>0$. To make use of this function, write first load("distrib").

The pdf is

$$f(x; a, b) = \cases{ \displaystyle{x^{a-1}(1-x)^{b-1} \over B(a,b)} & for $0 \le x \le 1$ \cr \cr 0 & otherwise }$$

$$f(x; a, b) = \cases{ \displaystyle{x^{a-1}(1-x)^{b-1} \over B(a,b)} & for $0 \le x \le 1$ \cr \cr 0 & otherwise }$$

pdf_binomial (x, n, p) — Function

Returns the value at x of the probability function of a ${\it Binomial}(n,p)$ random variable, with $0 \leq p \leq 1$ and $n$ a positive integer. To make use of this function, write first load("distrib").

The pdf is

$$f(x; n, p) = {n\choose x} (1-p)^{n-x}p^x$$

$$f(x; n, p) = {n\choose x} (1-p)^{n-x}p^x$$

pdf_cauchy (x, a, b) — Function

Returns the value at x of the density function of a ${\it Cauchy}(a,b)$ random variable, with $b>0$. To make use of this function, write first load("distrib").

The pdf is

$$f(x; a, b) = {b\over \pi\left((x-a)^2+b^2\right)}$$

$$f(x; a, b) = {b\over \pi\left((x-a)^2+b^2\right)}$$

pdf_chi2 (x, n) — Function

Returns the value at x of the density function of a Chi-square random variable $\chi^2(n)$ , with $n>0$. The $\chi^2(n)$ random variable is equivalent to the $\Gamma\left(n/2,2\right)$ .

The pdf is

$$f(x; n) = \cases{ \displaystyle{x^{n/2-1} e^{-x/2} \over 2^{n/2} \Gamma\left(\displaystyle{n\over 2}\right)} & for $x > 0$ \cr \cr 0 & otherwise }$$

$$f(x; n) = \cases{ \displaystyle{x^{n/2-1} e^{-x/2} \over 2^{n/2} \Gamma\left(\displaystyle{n\over 2}\right)} & for $x

0$ \cr \cr 0 & otherwise }$$

(%i1) load ("distrib")$

(%i2) pdf_chi2(x,n);
                    - x/2  n/2 - 1
                  %e      x        unit_step(x)
(%o2)             -----------------------------
                           n/2       n
                          2    gamma(-)
                                     2

pdf_continuous_uniform (x, a, b) — Function

Returns the value at x of the density function of a ${\it ContinuousUniform}(a,b)$ random variable, with $a \lt b.$ To make use of this function, write first load("distrib").

The pdf

$$f(x; a, b) = \cases{ \displaystyle{1\over b-a} & for $0 \le x \le 1$ \cr \cr 0 & otherwise }$$

$$f(x; a, b) = \cases{ \displaystyle{1\over b-a} & for $0 \le x \le 1$ \cr \cr 0 & otherwise }$$

and is 0 otherwise.

pdf_discrete_uniform (x, n) — Function

Returns the value at x of the probability function of a ${\it DiscreteUniform}(n)$ random variable, with $n$ a strictly positive integer. To make use of this function, write first load("distrib").

The pdf is

$$f(x,n) = {1\over n}$$

$$f(x,n) = {1\over n}$$

pdf_exp (x, m) — Function

Returns the value at x of the density function of an ${\it Exponential}(m)$ random variable, with $m>0$.

The ${\it Exponential}(m)$ random variable is equivalent to the ${\it Weibull}(1,1/m)$ .

The pdf is

$$f(x; m) = \cases{ me^{-mx} & for $x \ge 0$ \cr 0 & otherwise }$$

$$f(x; m) = \cases{ me^{-mx} & for $x \ge 0$ \cr 0 & otherwise }$$

(%i1) load ("distrib")$

(%i2) pdf_exp(x,m);
                       - m x
(%o2)                %e      m unit_step(x)

pdf_f (x, m, n) — Function

Returns the value at x of the density function of a F random variable $F(m,n)$, with $m,n>0$. To make use of this function, write first load("distrib").

The pdf is

$$f(x; m, n) = \cases{ B\left(\displaystyle{m\over 2}, \displaystyle{n\over 2}\right)^{-1} \left(\displaystyle{m\over n}\right)^{m/ 2} x^{m/2-1} \left(1 + \displaystyle{m\over n}x\right)^{-\left(n+m\right)/2} & $x > 0$ \cr \cr 0 & otherwise }$$

$$f(x; m, n) = \cases{ B\left(\displaystyle{m\over 2}, \displaystyle{n\over 2}\right)^{-1} \left(\displaystyle{m\over n}\right)^{m/ 2} x^{m/2-1} \left(1 + \displaystyle{m\over n}x\right)^{-\left(n+m\right)/2} & $x > 0$ \cr \cr 0 & otherwise }$$

pdf_gamma (x, a, b) — Function

Returns the value at x of the density function of a $\Gamma\left(a,b\right)$ random variable, with $a,b>0$. To make use of this function, write first load("distrib").

The shape parameter is $a$, and the scale parameter is $b$.

The pdf is

$$f(x; a, b) = {x^{a-1}e^{-x/b}\over b^a \Gamma(a)}$$

$$f(x; a, b) = {x^{a-1}e^{-x/b}\over b^a \Gamma(a)}$$

pdf_general_finite_discrete (x, v) — Function

Returns the value at x of the probability function of a general finite discrete random variable, with vector probabilities $v$, such that $Pr(X=i) = v_i$. Vector $v$ can be a list of nonnegative expressions whose components will be normalized to get a vector of probabilities. To make use of this function, write first load("distrib").

Note that $i=1$ corresponds to the first element of $v$.

(%i1) load ("distrib")$

(%i2) pdf_general_finite_discrete(2, [1/7, 4/7, 2/7]);
                                4
(%o2)                           -
                                7


(%i3) pdf_general_finite_discrete(2, [1, 4, 2]);
                                4
(%o3)                           -
                                7

pdf_geometric (x, p) — Function

Returns the value at x of the probability function of a ${\it Geometric}(p)$ random variable, with $0 < p \leq 1$

The pdf is

$$f(x; p) = p(1-p)^x$$

$$f(x; p) = p(1-p)^x$$

This is interpreted as the probability of $x$ failures before the first success.

load("distrib") loads this function.

pdf_gumbel (x, a, b) — Function

Returns the value at x of the density function of a ${\it Gumbel}(a,b)$ random variable, with $b>0$. To make use of this function, write first load("distrib").

The pdf is

$$f(x; a, b) = {1\over b} \exp\left[{a-x\over b} - \exp\left({a-x\over b}\right)\right]$$

$$f(x; a, b) = {1\over b} \exp\left[{a-x\over b} - \exp\left({a-x\over b}\right)\right]$$

pdf_hypergeometric (x, n_1, n_2, n) — Function

Returns the value at x of the probability function of a ${\it Hypergeometric}(n1,n2,n)$ random variable, with $n_1$, $n_2$ and $n$ non negative integers and $n\leq n_1+n_2$. Being $n_1$ the number of objects of class A, $n_2$ the number of objects of class B, and $n$ the size of the sample without replacement, this function returns the probability of event “exactly x objects are of class A”.

To make use of this function, write first load("distrib").

The pdf is

$$f(x; n_1, n_2, n) = {\displaystyle{n_1\choose x} {n_2 \choose n-x} \over \displaystyle{n_2+n_1 \choose n}}$$

$$f(x; n_1, n_2, n) = {\displaystyle{n_1\choose x} {n_2 \choose n-x} \over \displaystyle{n_2+n_1 \choose n}}$$

pdf_laplace (x, a, b) — Function

Returns the value at x of the density function of a ${\it Laplace}(a,b)$ random variable, with $b>0$. To make use of this function, write first load("distrib").

Here, $a$ is the location parameter (or mean), and $b$ is the scale parameter, related to the variance.

The pdf is

$$f(x; a, b) = {1\over 2b}\exp\left(-{|x-a|\over b}\right)$$

$$f(x; a, b) = {1\over 2b}\exp\left(-{|x-a|\over b}\right)$$

pdf_logistic (x, a, b) — Function

Returns the value at x of the density function of a ${\it Logistic}(a,b)$ random variable , with $b>0$. To make use of this function, write first load("distrib").

$a$ is the location parameter and $b$ is the scale parameter.

The pdf is

$$f(x; a, b) = {e^{-(x-a)/b} \over b\left(1 + e^{-(x-a)/b}\right)^2}$$

$$f(x; a, b) = {e^{-(x-a)/b} \over b\left(1 + e^{-(x-a)/b}\right)^2}$$

pdf_lognormal (x, m, s) — Function

Returns the value at x of the density function of a ${\it Lognormal}(m,s)$ random variable, with $s>0$. To make use of this function, write first load("distrib").

The pdf is

$$f(x; m, s) = \cases{ \displaystyle{1\over x s \sqrt{2\pi}} \exp\left(-\displaystyle{\left(\log x - m\right)^2\over 2s^2}\right) & for $x \ge 0$ \cr \cr 0 & for $x < 0$ }$$

$$f(x; m, s) = \cases{ \displaystyle{1\over x s \sqrt{2\pi}} \exp\left(-\displaystyle{\left(\log x - m\right)^2\over 2s^2}\right) & for $x \ge 0$ \cr \cr 0 & for $x < 0$ }$$

pdf_negative_binomial (x, n, p) — Function

Returns the value at x of the probability function of a ${\it NegativeBinomial}(n,p)$ random variable, with $0 < p \leq 1$ and $n$ a positive number. To make use of this function, write first load("distrib").

The pdf is

$$f(x; n, p) = {x+n-1 \choose n-1} (1-p)^xp^n$$

$$f(x; n, p) = {x+n-1 \choose n-1} (1-p)^xp^n$$

pdf_noncentral_chi2 (x, n, ncp) — Function

Returns the value at $x$ of the density function of a noncentral $\chi^2$ random variable m4_noncentral_chi2(n,ncp) , with $n>0$ and noncentrality parameter $ncp \ge 0.$ To make use of this function, write first load("distrib").

For $x < 0$, the pdf is 0, and for $x \ge 0$ the pdf is

$$f(x; n, \lambda) = {1\over 2}e^{-(x+\lambda)/2} \left(x\over \lambda\right)^{n/4-1/2}I_{{n\over 2} - 1}\left(\sqrt{n \lambda}\right)$$

$$f(x; n, \lambda) = {1\over 2}e^{-(x+\lambda)/2} \left(x\over \lambda\right)^{n/4-1/2}I_{{n\over 2} - 1}\left(\sqrt{n \lambda}\right) $$

pdf_noncentral_student_t (x, n, ncp) — Function

Returns the value at x of the density function of a noncentral Student random variable ${\it nc_t}(n, ncp)$ , with $n>0$ degrees of freedom and noncentrality parameter $ncp$. To make use of this function, write first load("distrib").

The pdf is

$$f(x; n, \mu) = \left[\sqrt{n} B\left({1\over 2}, {n\over 2}\right)\right]^{-1}\left(1+{x^2\over n}\right)^{-{(n+1)/2}} e^{-\mu^2/ 2} \bigg[A_n(x; \mu) + B_n(x; \mu)\bigg]$$

$$f(x; n, \mu) = \left[\sqrt{n} B\left({1\over 2}, {n\over 2}\right)\right]^{-1}\left(1+{x^2\over n}\right)^{-{(n+1)/2}} e^{-\mu^2/ 2} \bigg[A_n(x; \mu) + B_n(x; \mu)\bigg]$$

where

$$\eqalign{ A_n(x;\mu) &= {}_1F_1\left({n+1\over 2}; {1\over 2}; {\mu^2 x^2\over 2\left(x^2+n\right)}\right) \cr B_n(x;\mu) &= {\sqrt{2}\mu x \over \sqrt{x^2+n}} {\Gamma\left({n\over 2} + 1\right)\over \Gamma\left({n+1\over 2}\right)}; {}_1F_1\left({n\over 2} + 1; {3\over 2}; {\mu^2 x^2\over 2\left(x^2+n\right)}\right) }$$

$$\eqalign{ A_n(x;\mu) &= {}_1F_1\left({n+1\over 2}; {1\over 2}; {\mu^2 x^2\over 2\left(x^2+n\right)}\right) \cr B_n(x;\mu) &= {\sqrt{2}\mu x \over \sqrt{x^2+n}} {\Gamma\left({n\over 2} + 1\right)\over \Gamma\left({n+1\over 2}\right)}; {}_1F_1\left({n\over 2} + 1; {3\over 2}; {\mu^2 x^2\over 2\left(x^2+n\right)}\right) }$$

and $\mu$ is the non-centrality parameter $ncp$.

Sometimes an extra work is necessary to get the final result.

(%i1) load ("distrib")$

(%i2) expand(pdf_noncentral_student_t(3,5,0.1));
rat: replaced 0.018898223650461364 by 15934951/843198350 = 0.018898223650461364

rat: replaced -8.734356480209641 by -294697965/33740089 = -8.734356480209641

rat: replaced 4.136255165816327 by 51033443/12338079 = 4.136255165816332

rat: replaced 1.0806143216420299 by 49366521/45683756 = 1.0806143216420296

rat: replaced 0.0565127306411839 by 5608717/99246965 = 0.05651273064118384

rat: replaced -300.8069396896258 by -79782423/265228 = -300.80693968962555

rat: replaced 160.62691761849732 by 178374907/1110492 = 160.62691761849703
                            7/2                          7/2
      0.042964144174009046 5      1.3236503072892878e-6 5
(%o2) ------------------------- + --------------------------
         3/2   5/2                        sqrt(%pi)
        2    14    sqrt(%pi)
                                                              7/2
                                         1.94793720435093e-4 5
                                       + ------------------------
                                                   %pi


(%i3) float(%);
(%o3)                 0.020805931594056706

pdf_normal (x, m, s) — Function

Returns the value at x of the density function of a ${\it Normal}(m, s)$ random variable, with $s>0$. To make use of this function, write first load("distrib").

The pdf is

$$f(x; m, s) = {1\over s\sqrt{2\pi}} e^{\displaystyle -{(x-m)^2\over 2s^2}}$$

$$f(x; m, s) = {1\over s\sqrt{2\pi}} e^{\displaystyle -{(x-m)^2\over 2s^2}}$$

pdf_pareto (x, a, b) — Function

Returns the value at x of the density function of a ${\it Pareto}(a,b)$ random variable, with $a,b>0$. To make use of this function, write first load("distrib").

The pdf is

$$f(x; a, b) = \cases{ \displaystyle{a b^a \over x^{a+1}} & for $x \ge b$ \cr \cr 0 & for $x < b$ }$$

$$f(x; a, b) = \cases{ \displaystyle{a b^a \over x^{a+1}} & for $x \ge b$ \cr \cr 0 & for $x < b$ }$$

pdf_poisson (x, m) — Function

Returns the value at x of the probability function of a ${\it Poisson}(m)$ random variable, with $m>0$. To make use of this function, write first load("distrib").

The pdf is

$$f(x; m) = {m^x e^{-m}\over x!}$$

$$f(x; m) = {m^x e^{-m}\over x!}$$

pdf_rayleigh (x, b) — Function

Returns the value at x of the density function of a ${\it Rayleigh}(b)$ random variable, with $b>0$.

The ${\it Rayleigh}(b)$ random variable is equivalent to the ${\it Weibull}(2,1/b)$ .

The pdf is

$$f(x; b) = \cases{ 2b^2 x e^{-b^2 x^2} & for $x \ge 0$ \cr 0 & for $x < 0$ }$$

$$f(x; b) = \cases{ 2b^2 x e^{-b^2 x^2} & for $x \ge 0$ \cr 0 & for $x < 0$ }$$

(%i1) load ("distrib")$

(%i2) pdf_rayleigh(x,b);
                         2  2
                      - b  x   2
(%o2)             2 %e        b  x unit_step(x)

pdf_student_t (x, n) — Function

Returns the value at x of the density function of a Student random variable $t(n)$ , with $n>0$ degrees of freedom. To make use of this function, write first load("distrib").

The pdf is

$$f(x; n) = \left[\sqrt{n} B\left({1\over 2}, {n\over 2}\right)\right]^{-1} \left(1+{x^2\over n}\right)^{\displaystyle -{n+1\over 2}}$$

$$f(x; n) = \left[\sqrt{n} B\left({1\over 2}, {n\over 2}\right)\right]^{-1} \left(1+{x^2\over n}\right)^{\displaystyle -{n+1\over 2}}$$

pdf_weibull (x, a, b) — Function

Returns the value at x of the density function of a ${\it Weibull}(a,b)$ random variable, with $a,b>0$. To make use of this function, write first load("distrib").

The pdf is

$$f(x; a, b) = \cases{ \displaystyle{1\over b} \left({x\over b}\right)^{a-1} e^{-(x/b)^a} & for $x \ge 0$ \cr \cr 0 & for $x < 0$ }$$

$$f(x; a, b) = \cases{ \displaystyle{1\over b} \left({x\over b}\right)^{a-1} e^{-(x/b)^a} & for $x \ge 0$ \cr \cr 0 & for $x < 0$ }$$

quantile_bernoulli (q, p) — Function

Returns the q-quantile of a ${\it Bernoulli}(p)$ random variable, with $0 \leq p \leq 1$; in other words, this is the inverse of cdf_bernoulli. Argument q must be an element of $[0,1]$. To make use of this function, write first load("distrib").

quantile_beta (q, a, b) — Function

Returns the q-quantile of a ${\it Beta}(a,b)$ random variable, with $a,b>0$; in other words, this is the inverse of cdf_beta. Argument q must be an element of $[0,1]$. To make use of this function, write first load("distrib").

quantile_binomial (q, n, p) — Function

Returns the q-quantile of a ${\it Binomial}(n,p)$ random variable, with $0 \leq p \leq 1$ and $n$ a positive integer; in other words, this is the inverse of cdf_binomial. Argument q must be an element of $[0,1]$. To make use of this function, write first load("distrib").

quantile_cauchy (q, a, b) — Function

Returns the q-quantile of a ${\it Cauchy}(a,b)$ random variable, with $b>0$; in other words, this is the inverse of cdf_cauchy. Argument q must be an element of $[0,1]$. To make use of this function, write first load("distrib").

quantile_chi2 (q, n) — Function

Returns the q-quantile of a Chi-square random variable $\chi^2(n)$ , with $n>0$; in other words, this is the inverse of cdf_chi2. Argument q must be an element of $[0,1]$.

This function has no closed form and it is numerically computed.

(%i1) load ("distrib")$

(%i2) quantile_chi2(0.99,9);
(%o2)                   21.66599433346194

quantile_continuous_uniform (q, a, b) — Function

Returns the q-quantile of a ${\it ContinuousUniform}(a,b)$ random variable, with $a \lt b$ ; in other words, this is the inverse of cdf_continuous_uniform. Argument q must be an element of $[0,1]$. To make use of this function, write first load("distrib").

quantile_discrete_uniform (q, n) — Function

Returns the q-quantile of a ${\it DiscreteUniform}(n)$ random variable, with $n$ a strictly positive integer; in other words, this is the inverse of cdf_discrete_uniform. Argument q must be an element of $[0,1]$. To make use of this function, write first load("distrib").

quantile_exp (q, m) — Function

Returns the q-quantile of an ${\it Exponential}(m)$ random variable, with $m>0$; in other words, this is the inverse of cdf_exp. Argument q must be an element of $[0,1]$.

The ${\it Exponential}(m)$ random variable is equivalent to the ${\it Weibull}(1,1/m)$ .

(%i1) load ("distrib")$

(%i2) quantile_exp(0.56,5);
(%o2)                  0.1641961104139661


(%i3) quantile_exp(0.56,m);
                       0.8209805520698303
(%o3)                  ------------------
                               m

quantile_f (q, m, n) — Function

Returns the q-quantile of a F random variable $F(m,n)$, with $m,n>0$; in other words, this is the inverse of cdf_f. Argument q must be an element of $[0,1]$.

(%i1) load ("distrib")$

(%i2) quantile_f(2/5,sqrt(3),5);
(%o2)                  0.5189478385736904

quantile_gamma (q, a, b) — Function

Returns the q-quantile of a $\Gamma\left(a,b\right)$ random variable, with $a,b>0$; in other words, this is the inverse of cdf_gamma. Argument q must be an element of $[0,1]$. To make use of this function, write first load("distrib").

quantile_general_finite_discrete (q, v) — Function

Returns the q-quantile of a general finite discrete random variable, with vector probabilities $v$.

See pdf_005fgeneral_005ffinite_005fdiscrete for more details.

See also: pdf_general_finite_discrete.

quantile_geometric (q, p) — Function

Returns the q-quantile of a ${\it Geometric}(p)$ random variable, with $0 \lt p \le 1$ ; in other words, this is the inverse of cdf_geometric. Argument q must be an element of $[0,1]$.

The probability from which the quantile is derived is defined as $p (1 - p)^x$. This is interpreted as the probability of $x$ failures before the first success.

load("distrib") loads this function.

quantile_gumbel (q, a, b) — Function

Returns the q-quantile of a ${\it Gumbel}(a,b)$ random variable, with $b>0$; in other words, this is the inverse of cdf_gumbel. Argument q must be an element of $[0,1]$. To make use of this function, write first load("distrib").

quantile_hypergeometric (q, n1, n2, n) — Function

Returns the q-quantile of a ${\it Hypergeometric}(n1,n2,n)$ random variable, with n1, n2 and n non negative integers and $n\leq n1+n2$; in other words, this is the inverse of cdf_hypergeometric. Argument q must be an element of $[0,1]$. To make use of this function, write first load("distrib").

quantile_laplace (q, a, b) — Function

Returns the q-quantile of a ${\it Laplace}(a,b)$ random variable, with $b>0$; in other words, this is the inverse of cdf_laplace. Argument q must be an element of $[0,1]$. To make use of this function, write first load("distrib").

quantile_logistic (q, a, b) — Function

Returns the q-quantile of a ${\it Logistic}(a,b)$ random variable , with $b>0$; in other words, this is the inverse of cdf_logistic. Argument q must be an element of $[0,1]$. To make use of this function, write first load("distrib").

quantile_lognormal (q, m, s) — Function

Returns the q-quantile of a ${\it Lognormal}(m,s)$ random variable, with $s>0$; in other words, this is the inverse of cdf_lognormal. Argument q must be an element of $[0,1]$. To make use of this function, write first load("distrib").

(%i1) load ("distrib")$

(%i2) quantile_lognormal(95/100,0,1);
                     sqrt(2) inverse_erf(9/10)
(%o2)              %e


(%i3) float(%);
(%o3)                   5.180251602233015

quantile_negative_binomial (q, n, p) — Function

Returns the q-quantile of a ${\it NegativeBinomial}(n,p)$ random variable, with $0 < p \leq 1$ and $n$ a positive number; in other words, this is the inverse of cdf_negative_binomial. Argument q must be an element of $[0,1]$. To make use of this function, write first load("distrib").

quantile_noncentral_chi2 (q, n, ncp) — Function

Returns the q-quantile of a noncentral Chi-square random variable m4_noncentral_chi2(n,ncp) , with $n>0$ and noncentrality parameter $ncp \ge 0$ ; in other words, this is the inverse of cdf_noncentral_chi2. Argument q must be an element of $[0,1]$.

This function has no closed form and it is numerically computed.

quantile_noncentral_student_t (q, n, ncp) — Function

Returns the q-quantile of a noncentral Student random variable ${\it nc_t}(n, ncp)$ , with $n>0$ degrees of freedom and noncentrality parameter $ncp$; in other words, this is the inverse of cdf_noncentral_student_t. Argument q must be an element of $[0,1]$. To make use of this function, write first load("distrib").

quantile_normal (q, m, s) — Function

Returns the q-quantile of a ${\it Normal}(m, s)$ random variable, with $s>0$; in other words, this is the inverse of cdf_005fnormal. Argument q must be an element of $[0,1]$. To make use of this function, write first load("distrib").

(%i1) load ("distrib")$

(%i2) quantile_normal(95/100,0,1);
                                         9
(%o2)                sqrt(2) inverse_erf(--)
                                         10


(%i3) float(%);
(%o3)                  1.6448536269514724

See also: cdf_normal.

quantile_pareto (q, a, b) — Function

Returns the q-quantile of a ${\it Pareto}(a,b)$ random variable, with $a,b>0$; in other words, this is the inverse of cdf_pareto. Argument q must be an element of $[0,1]$. To make use of this function, write first load("distrib").

quantile_poisson (q, m) — Function

Returns the q-quantile of a ${\it Poisson}(m)$ random variable, with $m>0$; in other words, this is the inverse of cdf_poisson. Argument q must be an element of $[0,1]$. To make use of this function, write first load("distrib").

quantile_rayleigh (q, b) — Function

Returns the q-quantile of a ${\it Rayleigh}(b)$ random variable, with $b>0$; in other words, this is the inverse of cdf_rayleigh. Argument q must be an element of $[0,1]$.

The ${\it Rayleigh}(b)$ random variable is equivalent to the ${\it Weibull}(2,1/b)$ .

(%i1) load ("distrib")$

(%i2) quantile_rayleigh(0.99,b);
                       2.1459660262893467
(%o2)                  ------------------
                               b

quantile_student_t (q, n) — Function

Returns the q-quantile of a Student random variable $t(n)$ , with $n>0$; in other words, this is the inverse of cdf_student_t. Argument q must be an element of $[0,1]$. To make use of this function, write first load("distrib").

quantile_weibull (q, a, b) — Function

Returns the q-quantile of a ${\it Weibull}(a,b)$ random variable, with $a,b>0$; in other words, this is the inverse of cdf_weibull. Argument q must be an element of $[0,1]$. To make use of this function, write first load("distrib").

random_bernoulli (p) — Function

Returns a ${\it Bernoulli}(p)$ random variate, with $0 \leq p \leq 1$. Calling random_bernoulli with a second argument n, a random sample of size n will be simulated.

This is a direct application of the random built-in Maxima function.

See also random. To make use of this function, write first load("distrib").

See also: random.

random_beta (a, b) — Function

Returns a ${\it Beta}(a,b)$ random variate, with $a,b>0$. Calling random_beta with a third argument n, a random sample of size n will be simulated.

The implemented algorithm is defined in Cheng, R.C.H. (1978). Generating Beta Variates with Nonintegral Shape Parameters. Communications of the ACM, 21:317-322

To make use of this function, write first load("distrib").

random_binomial (n, p) — Function

Returns a ${\it Binomial}(n,p)$ random variate, with $0 \leq p \leq 1$ and $n$ a positive integer. Calling random_binomial with a third argument m, a random sample of size m will be simulated.

The implemented algorithm is based on the one described in Kachitvichyanukul, V. and Schmeiser, B.W. (1988) Binomial Random Variate Generation. Communications of the ACM, 31, Feb., 216.

To make use of this function, write first load("distrib").

random_cauchy (a, b) — Function

Returns a ${\it Cauchy}(a,b)$ random variate, with $b>0$. Calling random_cauchy with a third argument n, a random sample of size n will be simulated.

The implemented algorithm is based on the general inverse method.

To make use of this function, write first load("distrib").

random_chi2 (n) — Function

Returns a Chi-square random variate $\chi^2(n)$ , with $n>0$. Calling random_chi2 with a second argument m, a random sample of size m will be simulated.

The simulation is based on the Ahrens-Cheng algorithm. See random_gamma for details.

To make use of this function, write first load("distrib").

random_continuous_uniform (a, b) — Function

Returns a ${\it ContinuousUniform}(a,b)$ random variate, with $a \lt b.$ Calling random_continuous_uniform with a third argument n, a random sample of size n will be simulated.

This is a direct application of the random built-in Maxima function.

See also random. To make use of this function, write first load("distrib").

See also: random.

random_discrete_uniform (n) — Function

Returns a ${\it DiscreteUniform}(n)$ random variate, with $n$ a strictly positive integer. Calling random_discrete_uniform with a second argument m, a random sample of size m will be simulated.

This is a direct application of the random built-in Maxima function.

See also random. To make use of this function, write first load("distrib").

See also: random.

random_exp (m) — Function

Returns an ${\it Exponential}(m)$ random variate, with $m>0$. Calling random_exp with a second argument k, a random sample of size k will be simulated.

The simulation algorithm is based on the general inverse method.

To make use of this function, write first load("distrib").

random_f (m, n) — Function

Returns a F random variate $F(m,n)$, with $m,n>0$. Calling random_f with a third argument k, a random sample of size k will be simulated.

The simulation algorithm is based on the fact that if X is a $Chi^2(m)$ random variable and $Y$ is a $\chi^2(n)$ random variable, then

$$F={{n X}\over{m Y}}$$

$$F={{n X}\over{m Y}}$$

is a F random variable with m and n degrees of freedom, $F(m,n)$.

To make use of this function, write first load("distrib").

random_gamma (a, b) — Function

Returns a $\Gamma\left(a,b\right)$ random variate, with $a,b>0$. Calling random_gamma with a third argument n, a random sample of size n will be simulated.

The implemented algorithm is a combination of two procedures, depending on the value of parameter a:

For $a \ge 1,$ Cheng, R.C.H. and Feast, G.M. (1979). Some simple gamma variate generators. Appl. Stat., 28, 3, 290-295.

For $0 \lt a \lt 1,$ Ahrens, J.H. and Dieter, U. (1974). Computer methods for sampling from gamma, , poisson and binomial distributions. Computing, 12, 223-246.

To make use of this function, write first load("distrib").

random_general_finite_discrete (v) — Function

Returns a general finite discrete random variate, with vector probabilities $v$. Calling random_general_finite_discrete with a second argument m, a random sample of size m will be simulated.

See pdf_005fgeneral_005ffinite_005fdiscrete for more details.

(%i1) load ("distrib")$

(%i2) random_general_finite_discrete([1,3,1,5]);
(%o2)                           4


(%i3) random_general_finite_discrete([1,3,1,5], 10);
(%o3)            [4, 4, 2, 4, 2, 2, 4, 2, 2, 4]

See also: pdf_general_finite_discrete.

random_geometric (p) — Function

random_geometric(p) returns one random sample from a ${\it Geometric}(p)$ distribution, with $0 \lt p \le 1.$

random_geometric(p, n) returns a list of n random samples.

The algorithm is based on simulation of Bernoulli trials.

The probability from which the random sample is derived is defined as $p (1 - p)^x$. This is interpreted as the probability of $x$ failures before the first success.

load("distrib") loads this function.

random_gumbel (a, b) — Function

Returns a ${\it Gumbel}(a,b)$ random variate, with $b>0$. Calling random_gumbel with a third argument n, a random sample of size n will be simulated.

The implemented algorithm is based on the general inverse method.

To make use of this function, write first load("distrib").

random_hypergeometric (n1, n2, n) — Function

Returns a ${\it Hypergeometric}(n1,n2,n)$ random variate, with n1, n2 and n non negative integers and $n \le n_1 + n_2.$ Calling random_hypergeometric with a fourth argument m, a random sample of size m will be simulated.

Algorithm described in Kachitvichyanukul, V., Schmeiser, B.W. (1985) Computer generation of hypergeometric random variates. Journal of Statistical Computation and Simulation 22, 127-145.

To make use of this function, write first load("distrib").

random_laplace (a, b) — Function

Returns a ${\it Laplace}(a,b)$ random variate, with $b>0$. Calling random_laplace with a third argument n, a random sample of size n will be simulated.

The implemented algorithm is based on the general inverse method.

To make use of this function, write first load("distrib").

random_logistic (a, b) — Function

Returns a ${\it Logistic}(a,b)$ random variate, with $b>0$. Calling random_logistic with a third argument n, a random sample of size n will be simulated.

The implemented algorithm is based on the general inverse method.

To make use of this function, write first load("distrib").

random_lognormal (m, s) — Function

Returns a ${\it Lognormal}(m,s)$ random variate, with $s>0$. Calling random_lognormal with a third argument n, a random sample of size n will be simulated.

Log-normal variates are simulated by means of random normal variates. See random_normal for details.

To make use of this function, write first load("distrib").

random_negative_binomial (n, p) — Function

Returns a ${\it NegativeBinomial}(n,p)$ random variate, with $0 < p \leq 1$ and $n$ a positive number. Calling random_negative_binomial with a third argument m, a random sample of size m will be simulated.

Algorithm described in Devroye, L. (1986) Non-Uniform Random Variate Generation. Springer Verlag, p. 480.

To make use of this function, write first load("distrib").

random_noncentral_chi2 (n, ncp) — Function

Returns a noncentral Chi-square random variate m4_noncentral_chi2(n,ncp) , with $n>0$ and noncentrality parameter $ncp \ge 0.$ Calling random_noncentral_chi2 with a third argument m, a random sample of size m will be simulated.

To make use of this function, write first load("distrib").

random_noncentral_student_t (n, ncp) — Function

Returns a noncentral Student random variate ${\it nc_t}(n, ncp)$ , with $n>0$. Calling random_noncentral_student_t with a third argument m, a random sample of size m will be simulated.

The implemented algorithm is based on the fact that if X is a normal random variable ${\it Normal}(ncp, 1)$ and $S^2$ is a $\chi^2$ random variable with n degrees of freedom, $\chi^2(n)$ , then

$$U={{X}\over{\sqrt{{S^2}\over{n}}}}$$

$$U={{X}\over{\sqrt{{S^2}\over{n}}}}$$

is a noncentral Student random variable with $n$ degrees of freedom and noncentrality parameter $ncp$, ${\it nc_t}(n, ncp)$ .

To make use of this function, write first load("distrib").

random_normal (m, s) — Function

Returns a ${\it Normal}(m, s)$ random variate, with $s>0$. Calling random_normal with a third argument n, a random sample of size n will be simulated.

This is an implementation of the Box-Mueller algorithm, as described in Knuth, D.E. (1981) Seminumerical Algorithms. The Art of Computer Programming. Addison-Wesley.

To make use of this function, write first load("distrib").

random_pareto (a, b) — Function

Returns a ${\it Pareto}(a,b)$ random variate, with $a>0,b>0$. Calling random_pareto with a third argument n, a random sample of size n will be simulated.

The implemented algorithm is based on the general inverse method.

To make use of this function, write first load("distrib").

random_poisson (m) — Function

Returns a ${\it Poisson}(m)$ random variate, with $m>0$. Calling random_poisson with a second argument n, a random sample of size n will be simulated.

The implemented algorithm is the one described in Ahrens, J.H. and Dieter, U. (1982) Computer Generation of Poisson Deviates From Modified Normal Distributions. ACM Trans. Math. Software, 8, 2, June,163-179.

To make use of this function, write first load("distrib").

random_rayleigh (b) — Function

Returns a ${\it Rayleigh}(b)$ random variate, with $b>0$. Calling random_rayleigh with a second argument n, a random sample of size n will be simulated.

The implemented algorithm is based on the general inverse method.

To make use of this function, write first load("distrib").

random_student_t (n) — Function

Returns a Student random variate $t(n)$ , with $n>0$. Calling random_student_t with a second argument m, a random sample of size m will be simulated.

The implemented algorithm is based on the fact that if $Z$ is a normal random variable ${\it Normal}(0, 1)$ and $S^2$ is a $\chi^2$ random variable with $n$ degrees of freedom, $\chi^2(n)$ , then

$$X={{Z}\over{\sqrt{{S^2}\over{n}}}}$$

$$X={{Z}\over{\sqrt{{S^2}\over{n}}}}$$

is a Student random variable with $n$ degrees of freedom, $t(n)$ .

To make use of this function, write first load("distrib").

random_weibull (a, b) — Function

Returns a ${\it Weibull}(a,b)$ random variate, with $a,b>0$. Calling random_weibull with a third argument n, a random sample of size n will be simulated.

The implemented algorithm is based on the general inverse method.

To make use of this function, write first load("distrib").

skewness_bernoulli (p) — Function

Returns the skewness coefficient of a ${\it Bernoulli}(p)$ random variable, with $0 \leq p \leq 1$.

The ${\it Bernoulli}(p)$ random variable is equivalent to the ${\it Binomial}(1,p)$ .

The skewness coefficient is

$$SK[X] = {1-2p \over \sqrt{p(1-p)}}$$

$$SK[X] = {1-2p \over \sqrt{p(1-p)}}$$

(%i1) load ("distrib")$

(%i2) skewness_bernoulli(p);
                             1 - 2 p
(%o2)                    ---------------
                         sqrt((1 - p) p)

skewness_beta (a, b) — Function

Returns the skewness coefficient of a ${\it Beta}(a,b)$ random variable, with $a,b>0$. To make use of this function, write first load("distrib").

The skewness coefficient is

$$SK[X] = {2(b-a)\sqrt{a+b+1} \over (a+b+2)\sqrt{ab}}$$

$$SK[X] = {2(b-a)\sqrt{a+b+1} \over (a+b+2)\sqrt{ab}}$$

skewness_binomial (n, p) — Function

Returns the skewness coefficient of a ${\it Binomial}(n,p)$ random variable, with $0 \leq p \leq 1$ and $n$ a positive integer. To make use of this function, write first load("distrib").

The skewness coefficient is

$$SK[X] = {1-2p\over \sqrt{np(1-p)}}$$

$$SK[X] = {1-2p\over \sqrt{np(1-p)}}$$

skewness_chi2 (n) — Function

Returns the skewness coefficient of a Chi-square random variable $\chi^2(n)$ , with $n>0$.

The $\chi^2(n)$ random variable is equivalent to the $\Gamma\left(n/2,2\right)$ .

The skewness coefficient is

$$SK[X] = \sqrt{8\over n}$$

$$SK[X] = \sqrt{8\over n}$$

(%i1) load ("distrib")$

(%i2) skewness_chi2(n);
                               3/2
                              2
(%o2)                        -------
                             sqrt(n)

skewness_continuous_uniform (a, b) — Function

Returns the skewness coefficient of a ${\it ContinuousUniform}(a,b)$ random variable, with $a \lt b.$ To make use of this function, write first load("distrib").

The skewness coefficient is

$$SK[X] = 0$$

$$SK[X] = 0$$

skewness_discrete_uniform (n) — Function

Returns the skewness coefficient of a ${\it DiscreteUniform}(n)$ random variable, with $n$ a strictly positive integer. To make use of this function, write first load("distrib").

The skewness coefficient is

$$SK[X] = 0$$

$$SK[X] = 0$$

skewness_exp (m) — Function

Returns the skewness coefficient of an ${\it Exponential}(m)$ random variable, with $m>0$.

The ${\it Exponential}(m)$ random variable is equivalent to the ${\it Weibull}(1,1/m)$ .

The skewness coefficient is

$$SK[X] = 2$$

$$SK[X] = 2$$

(%i1) load ("distrib")$

(%i2) skewness_exp(m);
(%o2)                           2

skewness_f (m, n) — Function

Returns the skewness coefficient of a F random variable $F(m,n)$, with $m>0, n>6$. To make use of this function, write first load("distrib").

The skewness coefficient is

$$SK[X] = {(n+2m-2)\sqrt{8(n-4)} \over (n-6)\sqrt{m(n+m-2)}}$$

$$SK[X] = {(n+2m-2)\sqrt{8(n-4)} \over (n-6)\sqrt{m(n+m-2)}}$$

skewness_gamma (a, b) — Function

Returns the skewness coefficient of a $\Gamma\left(a,b\right)$ random variable, with $a,b>0$. To make use of this function, write first load("distrib").

The skewness coefficient is

$$SK[X] = {2\over \sqrt{a}}$$

$$SK[X] = {2\over \sqrt{a}}$$

skewness_general_finite_discrete (v) — Function

Returns the skewness coefficient of a general finite discrete random variable, with vector probabilities $v$.

See pdf_005fgeneral_005ffinite_005fdiscrete for more details.

See also: pdf_general_finite_discrete.

skewness_geometric (p) — Function

Returns the skewness coefficient of a ${\it Geometric}(p)$ random variable, with $0 < p \leq 1$.

The skewness coefficient is

$$SK[X] = {2-p \over \sqrt{1-p}}$$

$$SK[X] = {2-p \over \sqrt{1-p}}$$

load("distrib") loads this function.

skewness_gumbel (a, b) — Function

Returns the skewness coefficient of a ${\it Gumbel}(a,b)$ random variable, with $b>0$.

The skewness coefficient is

$$SK[X] = {12\sqrt{6}\over \pi^3} \zeta(3)$$

$$SK[X] = {12\sqrt{6}\over \pi^3} \zeta(3)$$

(%i1) load ("distrib")$

(%i2) skewness_gumbel(a,b);
                            3/2
                         2 6    zeta(3)
(%o2)                    --------------
                                 3
                              %pi

where zeta stands for the Riemann’s zeta function.

skewness_hypergeometric (n_1, n_2, n) — Function

Returns the skewness coefficient of a ${\it Hypergeometric}(n1,n2,n)$ random variable, with $n_1$, $n_2$ and $n$ non negative integers and $n\leq n1+n2$. To make use of this function, write first load("distrib").

The skewness coefficient is

$$SK[X] = {(n_2-n_2)(n_1+n_2-2n)\over n_1+n_2-2} \sqrt{n_1+n_2-1 \over n n_1 n_2 (n_1+n_2-n)}$$

$$SK[X] = {(n_2-n_2)(n_1+n_2-2n)\over n_1+n_2-2} \sqrt{n_1+n_2-1 \over n n_1 n_2 (n_1+n_2-n)}$$

skewness_laplace (a, b) — Function

Returns the skewness coefficient of a ${\it Laplace}(a,b)$ random variable, with $b>0$. To make use of this function, write first load("distrib").

The skewness coefficient is

$$SK[X] = 0$$

$$SK[X] = 0$$

skewness_logistic (a, b) — Function

Returns the skewness coefficient of a ${\it Logistic}(a,b)$ random variable, with $b>0$. To make use of this function, write first load("distrib").

The skewness coefficient is

$$SK[X] = 0$$

$$SK[X] = 0$$

skewness_lognormal (m, s) — Function

Returns the skewness coefficient of a ${\it Lognormal}(m,s)$ random variable, with $s>0$. To make use of this function, write first load("distrib").

The skewness coefficient is

$$SK[X] = \left(\exp\left(s^2\right)+2\right)\sqrt{\exp\left(s^2\right)-1}$$

$$SK[X] = \left(\exp\left(s^2\right)+2\right)\sqrt{\exp\left(s^2\right)-1}$$

skewness_negative_binomial (n, p) — Function

Returns the skewness coefficient of a ${\it NegativeBinomial}(n,p)$ random variable, with $0 < p \leq 1$ and $n$ a positive number. To make use of this function, write first load("distrib").

The skewness coefficient is

$$SK[X] = {2-p \over \sqrt{n(1-p)}}$$

$$SK[X] = {2-p \over \sqrt{n(1-p)}}$$

skewness_noncentral_chi2 (n, ncp) — Function

Returns the skewness coefficient of a noncentral Chi-square random variable m4_noncentral_chi2(n,ncp) , with $n>0$ and noncentrality parameter $ncp \ge 0.$

The skewness coefficient is

$$SK[X] = {2^{3/2}(n+3\mu) \over (n+2\mu)^{3/2}}$$

$$SK[X] = {2^{3/2}(n+3\mu) \over (n+2\mu)^{3/2}}$$

where $\mu$ is the noncentrality parameter ncp.

skewness_noncentral_student_t (n, ncp) — Function

Returns the skewness coefficient of a noncentral Student random variable ${\it nc_t}(n, ncp)$ , with $n>3$ degrees of freedom and noncentrality parameter $ncp$. To make use of this function, write first load("distrib").

If $U$ is a non-central Student’s $t$ random variable with $n$ degrees of freedom and a noncentrality parameter $\mu,$ the skewness is

$$\eqalign{ SK[U] &= {\mu\sqrt{n},\Gamma\left({{n-1}\over{2}}\right) \over{\sqrt{2}\Gamma\left({{n }\over{2}}\right)\sigma^{3}}}\left({{n \left(2n+\mu^2-3\right)}\over{\left(n-3\right)\left(n-2\right)}} -2\sigma^2\right) \cr \sigma^2 &= {{n\left(\mu^2+1\right)}\over{n-2}}-{{n \mu^2, \Gamma\left({{n-1}\over{2}}\right)^2}\over{2\Gamma\left({{n }\over{2}}\right)^2}} }$$

$$\eqalign{ SK[U] &= {\mu\sqrt{n},\Gamma\left({{n-1}\over{2}}\right) \over{\sqrt{2}\Gamma\left({{n }\over{2}}\right)\sigma^{3}}}\left({{n \left(2n+\mu^2-3\right)}\over{\left(n-3\right)\left(n-2\right)}} -2\sigma^2\right) \cr \sigma^2 &= {{n\left(\mu^2+1\right)}\over{n-2}}-{{n \mu^2, \Gamma\left({{n-1}\over{2}}\right)^2}\over{2\Gamma\left({{n }\over{2}}\right)^2}} } $$

skewness_normal (m, s) — Function

Returns the skewness coefficient of a ${\it Normal}(m, s)$ random variable, with $s>0$. To make use of this function, write first load("distrib").

The skewness coefficient is

$$SK[X] = 0$$

$$SK[X] = 0$$

skewness_pareto (a, b) — Function

Returns the skewness coefficient of a ${\it Pareto}(a,b)$ random variable, with $a>3,b>0$. To make use of this function, write first load("distrib").

The skewness coefficient is

$$SK[X] = {2(a+1)\over a-3} \sqrt{a-2\over a}$$

$$SK[X] = {2(a+1)\over a-3} \sqrt{a-2\over a}$$

skewness_poisson (m) — Function

Returns the skewness coefficient of a ${\it Poisson}(m)$ random variable, with $m>0$. To make use of this function, write first load("distrib").

The skewness is

$$SK[X] = {1\over \sqrt{m}}$$

$$SK[X] = {1\over \sqrt{m}}$$

skewness_rayleigh (b) — Function

Returns the skewness coefficient of a ${\it Rayleigh}(b)$ random variable, with $b>0$.

The ${\it Rayleigh}(b)$ random variable is equivalent to the ${\it Weibull}(2,1/b)$ .

The skewness coefficient is

$$SK[X] = {2\sqrt{\pi}(\pi - 3)\over (4-\pi)^{3/2}}$$

$$SK[X] = {2\sqrt{\pi}(\pi - 3)\over (4-\pi)^{3/2}}$$

(%i1) load ("distrib")$

(%i2) skewness_rayleigh(b);
                         3/2
                      %pi      3 sqrt(%pi)
                      ------ - -----------
                        4           4
(%o2)                 --------------------
                               %pi 3/2
                          (1 - ---)
                                4

skewness_student_t (n) — Function

Returns the skewness coefficient of a Student random variable $t(n)$ , with $n>3$, which is always equal to 0. To make use of this function, write first load("distrib").

The skewness coefficient is

$$SK[X] = 0$$

$$SK[X] = 0$$

skewness_weibull (a, b) — Function

Returns the skewness coefficient of a ${\it Weibull}(a,b)$ random variable, with $a,b>0$. To make use of this function, write first load("distrib").

The skewness coefficient is

$$SK[X] = {\displaystyle\Gamma\left(1+{3\over a}\right) -3\Gamma\left(1+{1\over a}\right)\Gamma\left(1+{2\over a}\right)+2\Gamma\left(1+{1\over a}\right)^3 \over \displaystyle\left[\Gamma\left(1+{2\over a}\right)-\Gamma\left(1+{1\over a}\right)^2\right]^{3/2} }$$

$$SK[X] = {\displaystyle\Gamma\left(1+{3\over a}\right) -3\Gamma\left(1+{1\over a}\right)\Gamma\left(1+{2\over a}\right)+2\Gamma\left(1+{1\over a}\right)^3 \over \displaystyle\left[\Gamma\left(1+{2\over a}\right)-\Gamma\left(1+{1\over a}\right)^2\right]^{3/2} } $$

std_bernoulli (p) — Function

Returns the standard deviation of a ${\it Bernoulli}(p)$ random variable, with $0 \leq p \leq 1$.

The ${\it Bernoulli}(p)$ random variable is equivalent to the ${\it Binomial}(1,p)$ .

The standard deviation is

$$D[X] = \sqrt{p(1-p)}$$

$$D[X] = \sqrt{p(1-p)}$$

(%i1) load ("distrib")$

(%i2) std_bernoulli(p);
(%o2)                    sqrt((1 - p) p)

std_beta (a, b) — Function

Returns the standard deviation of a ${\it Beta}(a,b)$ random variable, with $a,b>0$. To make use of this function, write first load("distrib").

The standard deviation is

$$D[X] = {1\over a+b}\sqrt{ab\over a+b+1}$$

$$D[X] = {1\over a+b}\sqrt{ab\over a+b+1}$$

std_binomial (n, p) — Function

Returns the standard deviation of a ${\it Binomial}(n,p)$ random variable, with $0 \leq p \leq 1$ and $n$ a positive integer. To make use of this function, write first load("distrib").

The standard deviation is

$$D[X] = \sqrt{np(1-p)}$$

$$D[X] = \sqrt{np(1-p)}$$

std_chi2 (n) — Function

Returns the standard deviation of a Chi-square random variable $\chi^2(n)$ , with $n>0$.

The $\chi^2(n)$ random variable is equivalent to the $\Gamma\left(n/2,2\right)$ .

The standard deviation is

$$D[X] = \sqrt{2n}$$

$$D[X] = \sqrt{2n}$$

(%i1) load ("distrib")$

(%i2) std_chi2(n);
(%o2)                    sqrt(2) sqrt(n)

std_continuous_uniform (a, b) — Function

Returns the standard deviation of a ${\it ContinuousUniform}(a,b)$ random variable, with $a \lt b.$ To make use of this function, write first load("distrib").

The standard deviation is

$$D[X] = {b-a \over 2\sqrt{3}}$$

$$D[X] = {b-a \over 2\sqrt{3}}$$

std_discrete_uniform (n) — Function

Returns the standard deviation of a ${\it DiscreteUniform}(n)$ random variable, with $n$ a strictly positive integer. To make use of this function, write first load("distrib").

The standard deviation is

$$D[X] = {\sqrt{n^2-1} \over 2\sqrt{3}}$$

$$D[X] = {\sqrt{n^2-1} \over 2\sqrt{3}}$$

std_exp (m) — Function

Returns the standard deviation of an ${\it Exponential}(m)$ random variable, with $m>0$.

The ${\it Exponential}(m)$ random variable is equivalent to the ${\it Weibull}(1,1/m)$ .

The standard deviation is

$$D[X] = {1\over m}$$

$$D[X] = {1\over m}$$

(%i1) load ("distrib")$

(%i2) std_exp(m);
                                1
(%o2)                           -
                                m

std_f (m, n) — Function

Returns the standard deviation of a F random variable $F(m,n)$, with $m>0, n>4$. To make use of this function, write first load("distrib").

The standard deviation is

$$D[X] = {\sqrt{2}, n \over n-2} \sqrt{n+m-2\over m(n-4)}$$

$$D[X] = {\sqrt{2}, n \over n-2} \sqrt{n+m-2\over m(n-4)}$$

std_gamma (a, b) — Function

Returns the standard deviation of a $\Gamma\left(a,b\right)$ random variable, with $a,b>0$. To make use of this function, write first load("distrib").

The standard deviation is

$$D[X] = b\sqrt{a}$$

$$D[X] = b\sqrt{a}$$

std_general_finite_discrete (v) — Function

Returns the standard deviation of a general finite discrete random variable, with vector probabilities $v$.

See pdf_005fgeneral_005ffinite_005fdiscrete for more details.

See also: pdf_general_finite_discrete.

std_geometric (p) — Function

Returns the standard deviation of a ${\it Geometric}(p)$ random variable, with $0 < p \leq 1$.

$$D[X] = {\sqrt{1-p} \over p}$$

$$D[X] = {\sqrt{1-p} \over p}$$

load("distrib") loads this function.

std_gumbel (a, b) — Function

Returns the standard deviation of a ${\it Gumbel}(a,b)$ random variable, with $b>0$. To make use of this function, write first load("distrib").

The standard deviation is

$$D[X] = {\pi \over \sqrt{6}} b$$

$$D[X] = {\pi \over \sqrt{6}} b$$

std_hypergeometric (n_1, n_2, n) — Function

Returns the standard deviation of a ${\it Hypergeometric}(n_1,n_2,n)$ random variable, with $n_1$, $n_2$ and $n$ non negative integers and $n\leq n_1+n_2$. To make use of this function, write first load("distrib").

The standard deviation is

$$D[X] = {1\over n_1+n_2}\sqrt{n n_1 n_2 (n_1 + n_2 - n) \over n_1+n_2-1}$$

$$D[X] = {1\over n_1+n_2}\sqrt{n n_1 n_2 (n_1 + n_2 - n) \over n_1+n_2-1}$$

std_laplace (a, b) — Function

Returns the standard deviation of a ${\it Laplace}(a,b)$ random variable, with $b>0$. To make use of this function, write first load("distrib").

The standard deviation is

$$D[X] = \sqrt{2} b$$

$$D[X] = \sqrt{2} b$$

std_logistic (a, b) — Function

Returns the standard deviation of a ${\it Logistic}(a,b)$ random variable , with $b>0$. To make use of this function, write first load("distrib").

The standard deviation is

$$D[X] = {\pi b\over \sqrt{3}}$$

$$D[X] = {\pi b\over \sqrt{3}}$$

std_lognormal (m, s) — Function

Returns the standard deviation of a ${\it Lognormal}(m,s)$ random variable, with $s>0$. To make use of this function, write first load("distrib").

The standard deviation is

$$D[X] = \sqrt{\left(\exp\left(s^2\right) - 1\right)} \exp\left(m+{s^2\over 2}\right)$$

$$D[X] = \sqrt{\left(\exp\left(s^2\right) - 1\right)} \exp\left(m+{s^2\over 2}\right)$$

std_negative_binomial (n, p) — Function

Returns the standard deviation of a ${\it NegativeBinomial}(n,p)$ random variable, with $0 < p \leq 1$ and $n$ a positive number. To make use of this function, write first load("distrib").

The standard deviation is

$$D[X] = {\sqrt{n(1-p)}\over p}$$

$$D[X] = {\sqrt{n(1-p)}\over p}$$

std_noncentral_chi2 (n, ncp) — Function

Returns the standard deviation of a noncentral Chi-square random variable m4_noncentral_chi2(n,ncp) , with $n>0$ and noncentrality parameter $ncp \ge 0.$

The standard deviation is

$$D[X] = \sqrt{2(n+2\mu)}$$

$$D[X] = \sqrt{2(n+2\mu)}$$

where $\mu$ is the noncentrality parameter ncp.

std_noncentral_student_t (n, ncp) — Function

Returns the standard deviation of a noncentral Student random variable ${\it nc_t}(n, ncp)$ , with $n>2$ degrees of freedom and noncentrality parameter $ncp$. To make use of this function, write first load("distrib").

The standard deviation is

$$D[X] = \sqrt{{n(\mu^2+1)\over n-2} - {n\mu^2; \Gamma\left(\displaystyle{n-1\over 2}\right)^2 \over 2\Gamma\left(\displaystyle{n\over 2}\right)^2}}$$

$$$$

std_normal (m, s) — Function

Returns the standard deviation of a ${\it Normal}(m, s)$ random variable, with $s>0$, namely s. To make use of this function, write first load("distrib").

The standard deviation is

$$D[X] = s$$

$$D[X] = s$$

std_pareto (a, b) — Function

Returns the standard deviation of a ${\it Pareto}(a,b)$ random variable, with $a>2,b>0$. To make use of this function, write first load("distrib").

The standard deviation is

$$D[X] = {b\over a-1} \sqrt{a\over a-2}$$

$$D[X] = {b\over a-1} \sqrt{a\over a-2}$$

std_poisson (m) — Function

Returns the standard deviation of a ${\it Poisson}(m)$ random variable, with $m>0$. To make use of this function, write first load("distrib").

The standard deviation is

$$V[X] = \sqrt{m}$$

$$V[X] = \sqrt{m}$$

std_rayleigh (b) — Function

Returns the standard deviation of a ${\it Rayleigh}(b)$ random variable, with $b>0$.

The ${\it Rayleigh}(b)$ random variable is equivalent to the ${\it Weibull}(2,1/b)$ .

The standard deviation is

$$D[X] = {1\over b}\sqrt{\displaystyle 1 - {\pi\over 4}}$$

$$D[X] = {1\over b}\sqrt{\displaystyle 1 - {\pi\over 4}}$$

(%i1) load ("distrib")$

(%i2) std_rayleigh(b);
                                   %pi
                          sqrt(1 - ---)
                                    4
(%o2)                     -------------
                                b

std_student_t (n) — Function

Returns the standard deviation of a Student random variable $t(n)$ , with $n>2$. To make use of this function, write first load("distrib").

The standard deviation is

$$D[X] = \sqrt{\displaystyle{n\over n-2}}$$

$$D[X] = \sqrt{\displaystyle{n\over n-2}}$$

std_weibull (a, b) — Function

Returns the standard deviation of a ${\it Weibull}(a,b)$ random variable, with $a,b>0$. To make use of this function, write first load("distrib").

The variance is

$$D[X] = b\sqrt{\Gamma\left(1+{2\over a}\right) - \Gamma\left(1+{1\over a}\right)^2}$$

$$D[X] = b\sqrt{\Gamma\left(1+{2\over a}\right) - \Gamma\left(1+{1\over a}\right)^2}$$

var_bernoulli (p) — Function

Returns the variance of a ${\it Bernoulli}(p)$ random variable, with $0 \leq p \leq 1$.

The ${\it Bernoulli}(p)$ random variable is equivalent to the ${\it Binomial}(1,p)$ .

The variance is

$$V[X] = p(1-p)$$

$$V[X] = p(1-p)$$

(%i1) load ("distrib")$

(%i2) var_bernoulli(p);
(%o2)                       (1 - p) p

var_beta (a, b) — Function

Returns the variance of a ${\it Beta}(a,b)$ random variable, with $a,b>0$. To make use of this function, write first load("distrib").

The variance is

$$V[X] = {ab \over (a+b)^2(a+b+1)}$$

$$V[X] = {ab \over (a+b)^2(a+b+1)}$$

var_binomial (n, p) — Function

Returns the variance of a ${\it Binomial}(n,p)$ random variable, with $0 \leq p \leq 1$ and $n$ a positive integer. To make use of this function, write first load("distrib").

The variance is

$$V[X] = np(1-p)$$

$$V[X] = np(1-p)$$

var_chi2 (n) — Function

Returns the variance of a Chi-square random variable $\chi^2(n)$ , with $n>0$.

The $\chi^2(n)$ random variable is equivalent to the $\Gamma\left(n/2,2\right)$ .

The variance is

$$V[X] = 2n$$

$$V[X] = 2n$$

(%i1) load ("distrib")$

(%i2) var_chi2(n);
(%o2)                          2 n

var_continuous_uniform (a, b) — Function

Returns the variance of a ${\it ContinuousUniform}(a,b)$ random variable, with $a \lt b.$ To make use of this function, write first load("distrib").

The variance is

$$V[X] = {(b-a)^2\over 12}$$

$$V[X] = {(b-a)^2\over 12}$$

var_discrete_uniform (n) — Function

Returns the variance of a ${\it DiscreteUniform}(n)$ random variable, with $n$ a strictly positive integer. To make use of this function, write first load("distrib").

The variance is

$$V[X] = {n^2-1 \over 12}$$

$$V[X] = {n^2-1 \over 12}$$

var_exp (m) — Function

Returns the variance of an ${\it Exponential}(m)$ random variable, with $m>0$.

The ${\it Exponential}(m)$ random variable is equivalent to the ${\it Weibull}(1,1/m)$ .

The variance is

$$V[X] = {1\over m^2}$$

$$V[X] = {1\over m^2}$$

(%i1) load ("distrib")$

(%i2) var_exp(m);
                               1
(%o2)                          --
                                2
                               m

var_f (m, n) — Function

Returns the variance of a F random variable $F(m,n)$, with $m>0, n>4$. To make use of this function, write first load("distrib").

The variance is

$$V[X] = {2n^2(n+m-2) \over m(n-4)(n-2)^2}$$

$$V[X] = {2n^2(n+m-2) \over m(n-4)(n-2)^2}$$

var_gamma (a, b) — Function

Returns the variance of a $\Gamma\left(a,b\right)$ random variable, with $a,b>0$. To make use of this function, write first load("distrib").

The variance is

$$V[X] = ab^2$$

$$V[X] = ab^2$$

var_general_finite_discrete (v) — Function

Returns the variance of a general finite discrete random variable, with vector probabilities $v$.

See pdf_005fgeneral_005ffinite_005fdiscrete for more details.

See also: pdf_general_finite_discrete.

var_geometric (p) — Function

Returns the variance of a ${\it Geometric}(p)$ random variable, with $0 < p \leq 1$.

The variance is

$$V[X] = {1-p\over p^2}$$

$$V[X] = {1-p\over p^2}$$

load("distrib") loads this function.

var_gumbel (a, b) — Function

Returns the variance of a ${\it Gumbel}(a,b)$ random variable, with $b>0$. To make use of this function, write first load("distrib").

The variance is

$$V[X] = {\pi^2\over 6} b^2$$

$$V[X] = {\pi^2\over 6} b^2$$

var_hypergeometric (n1, n2, n) — Function

Returns the variance of a hypergeometric random variable ${\it Hypergeometric}(n_1,n_2,n)$ , with $n_1$, $n_2$ and $n$ non negative integers and $n \le n_1 + n_2.$ To make use of this function, write first load("distrib").

The variance is

$$V[X] = {n n_1 n_2 (n_1 + n_2 - n) \over (n_1 + n_2 - 1) (n_1 + n_2)^2}$$

$$V[X] = {n n_1 n_2 (n_1 + n_2 - n) \over (n_1 + n_2 - 1) (n_1 + n_2)^2}$$

var_laplace (a, b) — Function

Returns the variance of a ${\it Laplace}(a,b)$ random variable, with $b>0$. To make use of this function, write first load("distrib").

The variance is

$$V[X] = 2b^2$$

$$V[X] = 2b^2$$

var_logistic (a, b) — Function

Returns the variance of a ${\it Logistic}(a,b)$ random variable , with $b>0$. To make use of this function, write first load("distrib").

The variance is

$$V[X] = {\pi^2 b^2 \over 3}$$

$$V[X] = {\pi^2 b^2 \over 3}$$

var_lognormal (m, s) — Function

Returns the variance of a ${\it Lognormal}(m,s)$ random variable, with $s>0$. To make use of this function, write first load("distrib").

The variance is

$$V[X] = \left(\exp\left(s^2\right) - 1\right) \exp\left(2m+s^2\right)$$

$$V[X] = \left(\exp\left(s^2\right) - 1\right) \exp\left(2m+s^2\right)$$

var_negative_binomial (n, p) — Function

Returns the variance of a ${\it NegativeBinomial}(n,p)$ random variable, with $0 < p \leq 1$ and $n$ a positive number. To make use of this function, write first load("distrib").

The variance is

$$V[X] = {n(1-p)\over p^2}$$

$$V[X] = {n(1-p)\over p^2}$$

var_noncentral_chi2 (n, ncp) — Function

Returns the variance of a noncentral Chi-square random variable m4_noncentral_chi2(n,ncp) , with $n>0$ and noncentrality parameter $ncp \ge 0.$

The variance is

$$V[X] = 2(n+2\mu)$$

$$V[X] = 2(n+2\mu)$$

where $\mu$ is the noncentrality parameter ncp.

var_noncentral_student_t (n, ncp) — Function

Returns the variance of a noncentral Student random variable ${\it nc_t}(n, ncp)$ , with $n>2$ degrees of freedom and noncentrality parameter $ncp$. To make use of this function, write first load("distrib").

The variance is

$$V[X] = {n(\mu^2+1)\over n-2} - {n\mu^2; \Gamma\left(\displaystyle{n-1\over 2}\right)^2 \over 2\Gamma\left(\displaystyle{n\over 2}\right)^2}$$

$$V[X] = {n(\mu^2+1)\over n-2} - {n\mu^2; \Gamma\left(\displaystyle{n-1\over 2}\right)^2 \over 2\Gamma\left(\displaystyle{n\over 2}\right)^2}$$

where $\mu$ is the noncentrality parameter $ncp$.

var_normal (m, s) — Function

Returns the variance of a ${\it Normal}(m, s)$ random variable, with $s>0$. To make use of this function, write first load("distrib").

The variance is

$$V[X] = s^2$$

$$V[X] = s^2$$

var_pareto (a, b) — Function

Returns the variance of a ${\it Pareto}(a,b)$ random variable, with $a>2,b>0$. To make use of this function, write first load("distrib").

The variance is

$$V[X] = {ab^2\over (a-2)(a-1)^2}$$

$$V[X] = {ab^2\over (a-2)(a-1)^2}$$

var_poisson (m) — Function

Returns the variance of a ${\it Poisson}(m)$ random variable, with $m>0$. To make use of this function, write first load("distrib").

The variance is

$$V[X] = m$$

$$V[X] = m$$

var_rayleigh (b) — Function

Returns the variance of a ${\it Rayleigh}(b)$ random variable, with $b>0$.

The ${\it Rayleigh}(b)$ random variable is equivalent to the ${\it Weibull}(2,1/b)$ .

The variance is

$$V[X] = {1\over b^2}\left(1-{\pi \over 4}\right)$$

$$V[X] = {1\over b^2}\left(1-{\pi \over 4}\right)$$

(%i1) load ("distrib")$

(%i2) var_rayleigh(b);
                                 %pi
                             1 - ---
                                  4
(%o2)                        -------
                                2
                               b

var_student_t (n) — Function

Returns the variance of a Student random variable $t(n)$ , with $n>2$.

The variance is

$$V[X] = {n\over n-2}$$

$$V[X] = {n\over n-2}$$

(%i1) load ("distrib")$

(%i2) var_student_t(n);
                                n
(%o2)                         -----
                              n - 2

var_weibull (a, b) — Function

Returns the variance of a ${\it Weibull}(a,b)$ random variable, with $a,b>0$. To make use of this function, write first load("distrib").

The variance is

$$V[X] = b^2\left[\Gamma\left(1+{2\over a}\right) - \Gamma\left(1+{1\over a}\right)^2\right]$$

$$V[X] = b^2\left[\Gamma\left(1+{2\over a}\right) - \Gamma\left(1+{1\over a}\right)^2\right]$$