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41.1 Introduction to distrib | ||

41.2 Functions and Variables for continuous distributions | ||

41.3 Functions and Variables for discrete distributions |

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Package `distrib`

contains a set of functions for making probability
computations on both discrete and continuous univariate models.

What follows is a short reminder of basic probabilistic related definitions.

Let *f(x)* be the `density function` of an absolute continuous random
variable *X*. The `distribution function` is defined as

x / [ F(x) = I f(u) du ] / minf

which equals the probability `Pr(X <= x)`.

The `mean` value is a localization parameter and is defined as

inf / [ E[X] = I x f(x) dx ] / minf

The `variance` is a measure of variation,

inf / [ 2 V[X] = I f(x) (x - E[X]) dx ] / minf

which is a positive real number. The square root of the variance is the
`standard deviation`, *D[X]=sqrt(V[X])*, and it is another measure
of variation.

The `skewness coefficient` is a measure of non-symmetry,

inf / 1 [ 3 SK[X] = ----- I f(x) (x - E[X]) dx 3 ] D[X] / minf

And the `kurtosis coefficient` measures the peakedness of the distribution,

inf / 1 [ 4 KU[X] = ----- I f(x) (x - E[X]) dx - 3 4 ] D[X] / minf

If *X* is gaussian, *KU[X]=0*. In fact, both skewness and kurtosis
are shape parameters used to measure the non-gaussianity of a distribution.

If the random variable *X* is discrete, the density, or `probability`,
function *f(x)* takes positive values within certain countable set of
numbers *x_i*, and zero elsewhere. In this case, the distribution function
is

==== \ F(x) = > f(x ) / i ==== x <= x i

The mean, variance, standard deviation, skewness coefficient and kurtosis coefficient take the form

==== \ E[X] = > x f(x ) , / i i ==== x i

==== \ 2 V[X] = > f(x ) (x - E[X]) , / i i ==== x i

D[X] = sqrt(V[X]),

==== 1 \ 3 SK[X] = ------- > f(x ) (x - E[X]) D[X]^3 / i i ==== x i

and

==== 1 \ 4 KU[X] = ------- > f(x ) (x - E[X]) - 3 , D[X]^4 / i i ==== x i

respectively.

There is a naming convention in package `distrib`

. Every function name has
two parts, the first one makes reference to the function or parameter we want
to calculate,

Functions: Density function (pdf_*) Distribution function (cdf_*) Quantile (quantile_*) Mean (mean_*) Variance (var_*) Standard deviation (std_*) Skewness coefficient (skewness_*) Kurtosis coefficient (kurtosis_*) Random variate (random_*)

The second part is an explicit reference to the probabilistic model,

Continuous distributions: Normal (*normal) Student (*student_t) Chi^2 (*chi2) Noncentral Chi^2 (*noncentral_chi2) F (*f) Exponential (*exp) Lognormal (*lognormal) Gamma (*gamma) Beta (*beta) Continuous uniform (*continuous_uniform) Logistic (*logistic) Pareto (*pareto) Weibull (*weibull) Rayleigh (*rayleigh) Laplace (*laplace) Cauchy (*cauchy) Gumbel (*gumbel) Discrete distributions: Binomial (*binomial) Poisson (*poisson) Bernoulli (*bernoulli) Geometric (*geometric) Discrete uniform (*discrete_uniform) hypergeometric (*hypergeometric) Negative binomial (*negative_binomial) Finite discrete (*general_finite_discrete)

For example, `pdf_student_t(x,n)`

is the density function of the Student
distribution with `n` degrees of freedom, `std_pareto(a,b)`

is the
standard deviation of the Pareto distribution with parameters `a` and
`b` and `kurtosis_poisson(m)`

is the kurtosis coefficient of the
Poisson distribution with mean `m`.

In order to make use of package `distrib`

you need first to load it by
typing

(%i1) load(distrib)$

For comments, bugs or suggestions, please contact the author at
`'mario AT edu DOT xunta DOT es'`.

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__Function:__**pdf_normal***(*`x`,`m`,`s`)Returns the value at

`x`of the density function of a*Normal(m,s)*random variable, with*s>0*. To make use of this function, write first`load(distrib)`

.

__Function:__**cdf_normal***(*`x`,`m`,`s`)Returns the value at

`x`of the distribution function of a*Normal(m,s)*random variable, with*s>0*. This function is defined in terms of Maxima's built-in error function`erf`

.(%i1) load (distrib)$ (%i2) assume(s>0)$ cdf_normal(x,m,s); x - m erf(---------) sqrt(2) s 1 (%o3) -------------- + - 2 2

See also

`erf`

.

__Function:__**quantile_normal***(*`q`,`m`,`s`)Returns the

`q`-quantile of a*Normal(m,s)*random variable, with*s>0*; in other words, this is the inverse of`cdf_normal`

. 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) 1.644853626951472

__Function:__**mean_normal***(*`m`,`s`)Returns the mean of a

*Normal(m,s)*random variable, with*s>0*, namely`m`. To make use of this function, write first`load(distrib)`

.

__Function:__**var_normal***(*`m`,`s`)Returns the variance of a

*Normal(m,s)*random variable, with*s>0*, namely`s^2`. To make use of this function, write first`load(distrib)`

.

__Function:__**std_normal***(*`m`,`s`)Returns the standard deviation of a

*Normal(m,s)*random variable, with*s>0*, namely`s`. To make use of this function, write first`load(distrib)`

.

__Function:__**skewness_normal***(*`m`,`s`)Returns the skewness coefficient of a

*Normal(m,s)*random variable, with*s>0*, which is always equal to 0. To make use of this function, write first`load(distrib)`

.

__Function:__**kurtosis_normal***(*`m`,`s`)Returns the kurtosis coefficient of a

*Normal(m,s)*random variable, with*s>0*, which is always equal to 0. To make use of this function, write first`load(distrib)`

.

__Function:__**random_normal***(*`m`,`s`)__Function:__**random_normal***(*`m`,`s`,`n`)Returns a

*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)`

.

__Function:__**pdf_student_t***(*`x`,`n`)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)`

.

__Function:__**cdf_student_t***(*`x`,`n`)Returns the value at

`x`of the distribution function of a Student random variable*t(n)*, with*n>0*degrees of freedom.(%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) .6698450596140415

__Function:__**quantile_student_t***(*`q`,`n`)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)`

.

__Function:__**mean_student_t***(*`n`)Returns the mean of a Student random variable

*t(n)*, with*n>0*, which is always equal to 0. To make use of this function, write first`load(distrib)`

.

__Function:__**var_student_t***(*`n`)Returns the variance of a Student random variable

*t(n)*, with*n>2*.(%i1) load (distrib)$ (%i2) assume(n>2)$ var_student_t(n); n (%o3) ----- n - 2

__Function:__**std_student_t***(*`n`)Returns the standard deviation of a Student random variable

*t(n)*, with*n>2*. To make use of this function, write first`load(distrib)`

.

__Function:__**skewness_student_t***(*`n`)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)`

.

__Function:__**kurtosis_student_t***(*`n`)Returns the kurtosis coefficient of a Student random variable

*t(n)*, with*n>4*. To make use of this function, write first`load(distrib)`

.

__Function:__**random_student_t***(*`n`)__Function:__**random_student_t***(*`n`,`m`)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*N(0,1)*and*S^2*is a chi square random variable with`n`degrees of freedom,*Chi^2(n)*, thenZ X = ------------- / 2 \ 1/2 | S | | --- | \ n /

is a Student random variable with

`n`degrees of freedom,*t(n)*.To make use of this function, write first

`load(distrib)`

.

__Function:__**pdf_noncentral_student_t***(*`x`,`n`,`ncp`)Returns the value at

`x`of the density function of a noncentral Student random variable*nc_t(n,ncp)*, with*n>0*degrees of freedom and noncentrality parameter*ncp*. To make use of this function, write first`load(distrib)`

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

(%i1) load (distrib)$ (%i2) expand(pdf_noncentral_student_t(3,5,0.1)); .01370030107589574 sqrt(5) (%o2) -------------------------- sqrt(2) sqrt(14) sqrt(%pi) 1.654562884111515E-4 sqrt(5) + ---------------------------- sqrt(%pi) .02434921505438663 sqrt(5) + -------------------------- %pi (%i3) float(%); (%o3) .02080593159405669

__Function:__**cdf_noncentral_student_t***(*`x`,`n`,`ncp`)Returns the value at

`x`of the distribution function of a noncentral Student random variable*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 if the global variable`numer`

equals`true`

or at least one of the arguments is a float, otherwise it returns a nominal expression.(%i1) load (distrib)$ (%i2) cdf_noncentral_student_t(-2,5,-5); (%o2) cdf_noncentral_student_t(- 2, 5, - 5) (%i3) cdf_noncentral_student_t(-2.0,5,-5); (%o3) .9952030093319743

__Function:__**quantile_noncentral_student_t***(*`q`,`n`,`ncp`)Returns the

`q`-quantile of a noncentral Student random variable*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)`

.

__Function:__**mean_noncentral_student_t***(*`n`,`ncp`)Returns the mean of a noncentral Student random variable

*nc_t(n,ncp)*, with*n>1*degrees of freedom and noncentrality parameter*ncp*. To make use of this function, write first`load(distrib)`

.(%i1) load (distrib)$ (%i2) (assume(df>1), mean_noncentral_student_t(df,k)); df - 1 gamma(------) sqrt(df) k 2 (%o2) ------------------------ df sqrt(2) gamma(--) 2

__Function:__**var_noncentral_student_t***(*`n`,`ncp`)Returns the variance of a noncentral Student random variable

*nc_t(n,ncp)*, with*n>2*degrees of freedom and noncentrality parameter*ncp*. To make use of this function, write first`load(distrib)`

.

__Function:__**std_noncentral_student_t***(*`n`,`ncp`)Returns the standard deviation of a noncentral Student random variable

*nc_t(n,ncp)*, with*n>2*degrees of freedom and noncentrality parameter*ncp*. To make use of this function, write first`load(distrib)`

.

__Function:__**skewness_noncentral_student_t***(*`n`,`ncp`)Returns the skewness coefficient of a noncentral Student random variable

*nc_t(n,ncp)*, with*n>3*degrees of freedom and noncentrality parameter*ncp*. To make use of this function, write first`load(distrib)`

.

__Function:__**kurtosis_noncentral_student_t***(*`n`,`ncp`)Returns the kurtosis coefficient of a noncentral Student random variable

*nc_t(n,ncp)*, with*n>4*degrees of freedom and noncentrality parameter*ncp*. To make use of this function, write first`load(distrib)`

.

__Function:__**random_noncentral_student_t***(*`n`,`ncp`)__Function:__**random_noncentral_student_t***(*`n`,`ncp`,`m`)Returns a noncentral Student random variate

*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*N(ncp,1)*and*S^2*is a chi square random variable with`n`degrees of freedom,*Chi^2(n)*, thenX U = ------------- / 2 \ 1/2 | S | | --- | \ n /

is a noncentral Student random variable with

`n`degrees of freedom and noncentrality parameter*ncp*,*nc_t(n,ncp)*.To make use of this function, write first

`load(distrib)`

.

__Function:__**pdf_chi2***(*`x`,`n`)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(n/2,2)*, therefore when Maxima has not enough information to get the result, a noun form based on the gamma density is returned.(%i1) load (distrib)$ (%i2) pdf_chi2(x,n); n (%o2) pdf_gamma(x, -, 2) 2 (%i3) assume(x>0, n>0)$ pdf_chi2(x,n); n/2 - 1 - x/2 x %e (%o4) ---------------- n/2 n 2 gamma(-) 2

__Function:__**cdf_chi2***(*`x`,`n`)Returns the value at

`x`of the distribution function of a Chi-square random variable*Chi^2(n)*, with*n>0*.(%i1) load (distrib)$ (%i2) cdf_chi2(3,4); 3 (%o2) 1 - gamma_incomplete_regularized(2, -) 2 (%i3) float(%); (%o3) .4421745996289256

__Function:__**quantile_chi2***(*`q`,`n`)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 if the global variable

`numer`

equals`true`

, otherwise it returns a nominal expression based on the gamma quantile function, since the*Chi^2(n)*random variable is equivalent to the*Gamma(n/2,2)*.(%i1) load (distrib)$ (%i2) quantile_chi2(0.99,9); (%o2) 21.66599433346194 (%i3) quantile_chi2(0.99,n); n (%o3) quantile_gamma(0.99, -, 2) 2

__Function:__**mean_chi2***(*`n`)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(n/2,2)*, therefore when Maxima has not enough information to get the result, a noun form based on the gamma mean is returned.(%i1) load (distrib)$ (%i2) mean_chi2(n); n (%o2) mean_gamma(-, 2) 2 (%i3) assume(n>0)$ mean_chi2(n); (%o4) n

__Function:__**var_chi2***(*`n`)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(n/2,2)*, therefore when Maxima has not enough information to get the result, a noun form based on the gamma variance is returned.(%i1) load (distrib)$ (%i2) var_chi2(n); n (%o2) var_gamma(-, 2) 2 (%i3) assume(n>0)$ var_chi2(n); (%o4) 2 n

__Function:__**std_chi2***(*`n`)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(n/2,2)*, therefore when Maxima has not enough information to get the result, a noun form based on the gamma standard deviation is returned.(%i1) load (distrib)$ (%i2) std_chi2(n); n (%o2) std_gamma(-, 2) 2 (%i3) assume(n>0)$ std_chi2(n); (%o4) sqrt(2) sqrt(n)

__Function:__**skewness_chi2***(*`n`)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(n/2,2)*, therefore when Maxima has not enough information to get the result, a noun form based on the gamma skewness coefficient is returned.(%i1) load (distrib)$ (%i2) skewness_chi2(n); n (%o2) skewness_gamma(-, 2) 2 (%i3) assume(n>0)$ skewness_chi2(n); 2 sqrt(2) (%o4) --------- sqrt(n)

__Function:__**kurtosis_chi2***(*`n`)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(n/2,2)*, therefore when Maxima has not enough information to get the result, a noun form based on the gamma kurtosis coefficient is returned.(%i1) load (distrib)$ (%i2) kurtosis_chi2(n); n (%o2) kurtosis_gamma(-, 2) 2 (%i3) assume(n>0)$ kurtosis_chi2(n); 12 (%o4) -- n

__Function:__**random_chi2***(*`n`)__Function:__**random_chi2***(*`n`,`m`)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)`

.

__Function:__**pdf_noncentral_chi2***(*`x`,`n`,`ncp`)Returns the value at

`x`of the density function of a noncentral Chi-square random variable*nc_Chi^2(n,ncp)*, with*n>0*and noncentrality parameter*ncp>=0*. To make use of this function, write first`load(distrib)`

.

__Function:__**cdf_noncentral_chi2***(*`x`,`n`,`ncp`)Returns the value at

`x`of the distribution function of a noncentral Chi-square random variable*nc_Chi^2(n,ncp)*, with*n>0*and noncentrality parameter*ncp>=0*. To make use of this function, write first`load(distrib)`

.

__Function:__**quantile_noncentral_chi2***(*`q`,`n`,`ncp`)Returns the

`q`-quantile of a noncentral Chi-square random variable*nc_Chi^2(n,ncp)*, with*n>0*and noncentrality parameter*ncp>=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 if the global variable

`numer`

equals`true`

, otherwise it returns a nominal expression.

__Function:__**mean_noncentral_chi2***(*`n`,`ncp`)Returns the mean of a noncentral Chi-square random variable

*nc_Chi^2(n,ncp)*, with*n>0*and noncentrality parameter*ncp>=0*.

__Function:__**var_noncentral_chi2***(*`n`,`ncp`)Returns the variance of a noncentral Chi-square random variable

*nc_Chi^2(n,ncp)*, with*n>0*and noncentrality parameter*ncp>=0*.

__Function:__**std_noncentral_chi2***(*`n`,`ncp`)Returns the standard deviation of a noncentral Chi-square random variable

*nc_Chi^2(n,ncp)*, with*n>0*and noncentrality parameter*ncp>=0*.

__Function:__**skewness_noncentral_chi2***(*`n`,`ncp`)Returns the skewness coefficient of a noncentral Chi-square random variable

*nc_Chi^2(n,ncp)*, with*n>0*and noncentrality parameter*ncp>=0*.

__Function:__**kurtosis_noncentral_chi2***(*`n`,`ncp`)Returns the kurtosis coefficient of a noncentral Chi-square random variable

*nc_Chi^2(n,ncp)*, with*n>0*and noncentrality parameter*ncp>=0*.

__Function:__**random_noncentral_chi2***(*`n`,`ncp`)__Function:__**random_noncentral_chi2***(*`n`,`ncp`,`m`)Returns a noncentral Chi-square random variate

*nc_Chi^2(n,ncp)*, with*n>0*and noncentrality parameter*ncp>=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)`

.

__Function:__**pdf_f***(*`x`,`m`,`n`)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)`

.

__Function:__**cdf_f***(*`x`,`m`,`n`)Returns the value at

`x`of the distribution function of a F random variable*F(m,n)*, with*m,n>0*.(%i1) load (distrib)$ (%i2) cdf_f(2,3,9/4); 9 3 3 (%o2) 1 - beta_incomplete_regularized(-, -, --) 8 2 11 (%i3) float(%); (%o3) 0.66756728179008

__Function:__**quantile_f***(*`q`,`m`,`n`)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]*.This function has no closed form and it is numerically computed if the global variable

`numer`

equals`true`

, otherwise it returns a nominal expression.(%i1) load (distrib)$ (%i2) quantile_f(2/5,sqrt(3),5); 2 (%o2) quantile_f(-, sqrt(3), 5) 5 (%i3) %,numer; (%o3) 0.518947838573693

__Function:__**mean_f***(*`m`,`n`)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)`

.

__Function:__**var_f***(*`m`,`n`)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)`

.

__Function:__**std_f***(*`m`,`n`)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)`

.

__Function:__**skewness_f***(*`m`,`n`)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)`

.

__Function:__**kurtosis_f***(*`m`,`n`)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)`

.

__Function:__**random_f***(*`m`,`n`)__Function:__**random_f***(*`m`,`n`,`k`)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, thenn X F = --- 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)`

.

__Function:__**pdf_exp***(*`x`,`m`)Returns the value at

`x`of the density function of an*Exponential(m)*random variable, with*m>0*.The

*Exponential(m)*random variable is equivalent to the*Weibull(1,1/m)*, therefore when Maxima has not enough information to get the result, a noun form based on the Weibull density is returned.(%i1) load (distrib)$ (%i2) pdf_exp(x,m); 1 (%o2) pdf_weibull(x, 1, -) m (%i3) assume(x>0,m>0)$ pdf_exp(x,m); - m x (%o4) m %e

__Function:__**cdf_exp***(*`x`,`m`)Returns the value at

`x`of the distribution function of an*Exponential(m)*random variable, with*m>0*.The

*Exponential(m)*random variable is equivalent to the*Weibull(1,1/m)*, therefore when Maxima has not enough information to get the result, a noun form based on the Weibull distribution is returned.(%i1) load (distrib)$ (%i2) cdf_exp(x,m); 1 (%o2) cdf_weibull(x, 1, -) m (%i3) assume(x>0,m>0)$ cdf_exp(x,m); - m x (%o4) 1 - %e

__Function:__**quantile_exp***(*`q`,`m`)Returns the

`q`-quantile of an*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

*Exponential(m)*random variable is equivalent to the*Weibull(1,1/m)*, therefore when Maxima has not enough information to get the result, a noun form based on the Weibull quantile is returned.(%i1) load (distrib)$ (%i2) quantile_exp(0.56,5); (%o2) .1641961104139661 (%i3) quantile_exp(0.56,m); 1 (%o3) quantile_weibull(0.56, 1, -) m

__Function:__**mean_exp***(*`m`)Returns the mean of an

*Exponential(m)*random variable, with*m>0*.The

*Exponential(m)*random variable is equivalent to the*Weibull(1,1/m)*, therefore when Maxima has not enough information to get the result, a noun form based on the Weibull mean is returned.(%i1) load (distrib)$ (%i2) mean_exp(m); 1 (%o2) mean_weibull(1, -) m (%i3) assume(m>0)$ mean_exp(m); 1 (%o4) - m

__Function:__**var_exp***(*`m`)Returns the variance of an

*Exponential(m)*random variable, with*m>0*.The

*Exponential(m)*random variable is equivalent to the*Weibull(1,1/m)*, therefore when Maxima has not enough information to get the result, a noun form based on the Weibull variance is returned.(%i1) load (distrib)$ (%i2) var_exp(m); 1 (%o2) var_weibull(1, -) m (%i3) assume(m>0)$ var_exp(m); 1 (%o4) -- 2 m

__Function:__**std_exp***(*`m`)Returns the standard deviation of an

*Exponential(m)*random variable, with*m>0*.The

*Exponential(m)*random variable is equivalent to the*Weibull(1,1/m)*, therefore when Maxima has not enough information to get the result, a noun form based on the Weibull standard deviation is returned.(%i1) load (distrib)$ (%i2) std_exp(m); 1 (%o2) std_weibull(1, -) m (%i3) assume(m>0)$ std_exp(m); 1 (%o4) - m

__Function:__**skewness_exp***(*`m`)Returns the skewness coefficient of an

*Exponential(m)*random variable, with*m>0*.The

*Exponential(m)*random variable is equivalent to the*Weibull(1,1/m)*, therefore when Maxima has not enough information to get the result, a noun form based on the Weibull skewness coefficient is returned.(%i1) load (distrib)$ (%i2) skewness_exp(m); 1 (%o2) skewness_weibull(1, -) m (%i3) assume(m>0)$ skewness_exp(m); (%o4) 2

__Function:__**kurtosis_exp***(*`m`)Returns the kurtosis coefficient of an

*Exponential(m)*random variable, with*m>0*.The

*Exponential(m)*random variable is equivalent to the*Weibull(1,1/m)*, therefore when Maxima has not enough information to get the result, a noun form based on the Weibull kurtosis coefficient is returned.(%i1) load (distrib)$ (%i2) kurtosis_exp(m); 1 (%o2) kurtosis_weibull(1, -) m (%i3) assume(m>0)$ kurtosis_exp(m); (%o4) 6

__Function:__**random_exp***(*`m`)__Function:__**random_exp***(*`m`,`k`)Returns an

*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)`

.

__Function:__**pdf_lognormal***(*`x`,`m`,`s`)Returns the value at

`x`of the density function of a*Lognormal(m,s)*random variable, with*s>0*. To make use of this function, write first`load(distrib)`

.

__Function:__**cdf_lognormal***(*`x`,`m`,`s`)Returns the value at

`x`of the distribution function of a*Lognormal(m,s)*random variable, with*s>0*. This function is defined in terms of Maxima's built-in error function`erf`

.(%i1) load (distrib)$ (%i2) assume(x>0, s>0)$ cdf_lognormal(x,m,s); log(x) - m erf(----------) sqrt(2) s 1 (%o3) --------------- + - 2 2

See also

`erf`

.

__Function:__**quantile_lognormal***(*`q`,`m`,`s`)Returns the

`q`-quantile of a*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

__Function:__**mean_lognormal***(*`m`,`s`)Returns the mean of a

*Lognormal(m,s)*random variable, with*s>0*. To make use of this function, write first`load(distrib)`

.

__Function:__**var_lognormal***(*`m`,`s`)Returns the variance of a

*Lognormal(m,s)*random variable, with*s>0*. To make use of this function, write first`load(distrib)`

.

__Function:__**std_lognormal***(*`m`,`s`)Returns the standard deviation of a

*Lognormal(m,s)*random variable, with*s>0*. To make use of this function, write first`load(distrib)`

.

__Function:__**skewness_lognormal***(*`m`,`s`)Returns the skewness coefficient of a

*Lognormal(m,s)*random variable, with*s>0*. To make use of this function, write first`load(distrib)`

.

__Function:__**kurtosis_lognormal***(*`m`,`s`)Returns the kurtosis coefficient of a

*Lognormal(m,s)*random variable, with*s>0*. To make use of this function, write first`load(distrib)`

.

__Function:__**random_lognormal***(*`m`,`s`)__Function:__**random_lognormal***(*`m`,`s`,`n`)Returns a

*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)`

.

__Function:__**pdf_gamma***(*`x`,`a`,`b`)Returns the value at

`x`of the density function of a*Gamma(a,b)*random variable, with*a,b>0*. To make use of this function, write first`load(distrib)`

.

__Function:__**cdf_gamma***(*`x`,`a`,`b`)Returns the value at

`x`of the distribution function of a*Gamma(a,b)*random variable, with*a,b>0*.(%i1) load (distrib)$ (%i2) cdf_gamma(3,5,21); 1 (%o2) 1 - gamma_incomplete_regularized(5, -) 7 (%i3) float(%); (%o3) 4.402663157376807E-7

__Function:__**quantile_gamma***(*`q`,`a`,`b`)Returns the

`q`-quantile of a*Gamma(a,b)*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)`

.

__Function:__**mean_gamma***(*`a`,`b`)Returns the mean of a

*Gamma(a,b)*random variable, with*a,b>0*. To make use of this function, write first`load(distrib)`

.

__Function:__**var_gamma***(*`a`,`b`)Returns the variance of a

*Gamma(a,b)*random variable, with*a,b>0*. To make use of this function, write first`load(distrib)`

.

__Function:__**std_gamma***(*`a`,`b`)Returns the standard deviation of a

*Gamma(a,b)*random variable, with*a,b>0*. To make use of this function, write first`load(distrib)`

.

__Function:__**skewness_gamma***(*`a`,`b`)Returns the skewness coefficient of a

*Gamma(a,b)*random variable, with*a,b>0*. To make use of this function, write first`load(distrib)`

.

__Function:__**kurtosis_gamma***(*`a`,`b`)Returns the kurtosis coefficient of a

*Gamma(a,b)*random variable, with*a,b>0*. To make use of this function, write first`load(distrib)`

.

__Function:__**random_gamma***(*`a`,`b`)__Function:__**random_gamma***(*`a`,`b`,`n`)Returns a

*Gamma(a,b)*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>=1*, Cheng, R.C.H. and Feast, G.M. (1979).`Some simple gamma variate generators`. Appl. Stat., 28, 3, 290-295.For

*0<a<1*, Ahrens, J.H. and Dieter, U. (1974).`Computer methods for sampling from gamma, beta, poisson and binomial cdf_tributions`. Computing, 12, 223-246.To make use of this function, write first

`load(distrib)`

.

__Function:__**pdf_beta***(*`x`,`a`,`b`)Returns the value at

`x`of the density function of a*Beta(a,b)*random variable, with*a,b>0*. To make use of this function, write first`load(distrib)`

.

__Function:__**cdf_beta***(*`x`,`a`,`b`)Returns the value at

`x`of the distribution function of a*Beta(a,b)*random variable, with*a,b>0*.(%i1) load (distrib)$ (%i2) cdf_beta(1/3,15,2); 11 (%o2) -------- 14348907 (%i3) float(%); (%o3) 7.666089131388195E-7

__Function:__**quantile_beta***(*`q`,`a`,`b`)Returns the

`q`-quantile of a*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)`

.

__Function:__**mean_beta***(*`a`,`b`)Returns the mean of a

*Beta(a,b)*random variable, with*a,b>0*. To make use of this function, write first`load(distrib)`

.

__Function:__**var_beta***(*`a`,`b`)Returns the variance of a

*Beta(a,b)*random variable, with*a,b>0*. To make use of this function, write first`load(distrib)`

.

__Function:__**std_beta***(*`a`,`b`)Returns the standard deviation of a

*Beta(a,b)*random variable, with*a,b>0*. To make use of this function, write first`load(distrib)`

.

__Function:__**skewness_beta***(*`a`,`b`)Returns the skewness coefficient of a

*Beta(a,b)*random variable, with*a,b>0*. To make use of this function, write first`load(distrib)`

.

__Function:__**kurtosis_beta***(*`a`,`b`)Returns the kurtosis coefficient of a

*Beta(a,b)*random variable, with*a,b>0*. To make use of this function, write first`load(distrib)`

.

__Function:__**random_beta***(*`a`,`b`)__Function:__**random_beta***(*`a`,`b`,`n`)Returns a

*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, 21:317-322To make use of this function, write first

`load(distrib)`

.

__Function:__**pdf_continuous_uniform***(*`x`,`a`,`b`)Returns the value at

`x`of the density function of a*Continuous Uniform(a,b)*random variable, with*a<b*. To make use of this function, write first`load(distrib)`

.

__Function:__**cdf_continuous_uniform***(*`x`,`a`,`b`)Returns the value at

`x`of the distribution function of a*Continuous Uniform(a,b)*random variable, with*a<b*. To make use of this function, write first`load(distrib)`

.

__Function:__**quantile_continuous_uniform***(*`q`,`a`,`b`)Returns the

`q`-quantile of a*Continuous Uniform(a,b)*random variable, with*a<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)`

.

__Function:__**mean_continuous_uniform***(*`a`,`b`)Returns the mean of a

*Continuous Uniform(a,b)*random variable, with*a<b*. To make use of this function, write first`load(distrib)`

.

__Function:__**var_continuous_uniform***(*`a`,`b`)Returns the variance of a

*Continuous Uniform(a,b)*random variable, with*a<b*. To make use of this function, write first`load(distrib)`

.

__Function:__**std_continuous_uniform***(*`a`,`b`)Returns the standard deviation of a

*Continuous Uniform(a,b)*random variable, with*a<b*. To make use of this function, write first`load(distrib)`

.

__Function:__**skewness_continuous_uniform***(*`a`,`b`)Returns the skewness coefficient of a

*Continuous Uniform(a,b)*random variable, with*a<b*. To make use of this function, write first`load(distrib)`

.

__Function:__**kurtosis_continuous_uniform***(*`a`,`b`)Returns the kurtosis coefficient of a

*Continuous Uniform(a,b)*random variable, with*a<b*. To make use of this function, write first`load(distrib)`

.

__Function:__**random_continuous_uniform***(*`a`,`b`)__Function:__**random_continuous_uniform***(*`a`,`b`,`n`)Returns a

*Continuous Uniform(a,b)*random variate, with*a<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)`

.

__Function:__**pdf_logistic***(*`x`,`a`,`b`)Returns the value at

`x`of the density function of a*Logistic(a,b)*random variable, with*b>0*. To make use of this function, write first`load(distrib)`

.

__Function:__**cdf_logistic***(*`x`,`a`,`b`)Returns the value at

`x`of the distribution function of a*Logistic(a,b)*random variable, with*b>0*. To make use of this function, write first`load(distrib)`

.

__Function:__**quantile_logistic***(*`q`,`a`,`b`)Returns the

`q`-quantile of a*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)`

.

__Function:__**mean_logistic***(*`a`,`b`)Returns the mean of a

*Logistic(a,b)*random variable, with*b>0*. To make use of this function, write first`load(distrib)`

.

__Function:__**var_logistic***(*`a`,`b`)Returns the variance of a

*Logistic(a,b)*random variable, with*b>0*. To make use of this function, write first`load(distrib)`

.

__Function:__**std_logistic***(*`a`,`b`)Returns the standard deviation of a

*Logistic(a,b)*random variable, with*b>0*. To make use of this function, write first`load(distrib)`

.

__Function:__**skewness_logistic***(*`a`,`b`)Returns the skewness coefficient of a

*Logistic(a,b)*random variable, with*b>0*. To make use of this function, write first`load(distrib)`

.

__Function:__**kurtosis_logistic***(*`a`,`b`)Returns the kurtosis coefficient of a

*Logistic(a,b)*random variable, with*b>0*. To make use of this function, write first`load(distrib)`

.

__Function:__**random_logistic***(*`a`,`b`)__Function:__**random_logistic***(*`a`,`b`,`n`)Returns a

*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)`

.

__Function:__**pdf_pareto***(*`x`,`a`,`b`)Returns the value at

`x`of the density function of a*Pareto(a,b)*random variable, with*a,b>0*. To make use of this function, write first`load(distrib)`

.

__Function:__**cdf_pareto***(*`x`,`a`,`b`)Returns the value at

`x`of the distribution function of a*Pareto(a,b)*random variable, with*a,b>0*. To make use of this function, write first`load(distrib)`

.

__Function:__**quantile_pareto***(*`q`,`a`,`b`)Returns the

`q`-quantile of a*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)`

.

__Function:__**mean_pareto***(*`a`,`b`)Returns the mean of a

*Pareto(a,b)*random variable, with*a>1,b>0*. To make use of this function, write first`load(distrib)`

.

__Function:__**var_pareto***(*`a`,`b`)Returns the variance of a

*Pareto(a,b)*random variable, with*a>2,b>0*. To make use of this function, write first`load(distrib)`

.

__Function:__**std_pareto***(*`a`,`b`)Returns the standard deviation of a

*Pareto(a,b)*random variable, with*a>2,b>0*. To make use of this function, write first`load(distrib)`

.

__Function:__**skewness_pareto***(*`a`,`b`)Returns the skewness coefficient of a

*Pareto(a,b)*random variable, with*a>3,b>0*. To make use of this function, write first`load(distrib)`

.

__Function:__**kurtosis_pareto***(*`a`,`b`)Returns the kurtosis coefficient of a

*Pareto(a,b)*random variable, with*a>4,b>0*. To make use of this function, write first`load(distrib)`

.

__Function:__**random_pareto***(*`a`,`b`)__Function:__**random_pareto***(*`a`,`b`,`n`)Returns a

*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)`

.

__Function:__**pdf_weibull***(*`x`,`a`,`b`)Returns the value at

`x`of the density function of a*Weibull(a,b)*random variable, with*a,b>0*. To make use of this function, write first`load(distrib)`

.

__Function:__**cdf_weibull***(*`x`,`a`,`b`)Returns the value at

`x`of the distribution function of a*Weibull(a,b)*random variable, with*a,b>0*. To make use of this function, write first`load(distrib)`

.

__Function:__**quantile_weibull***(*`q`,`a`,`b`)Returns the

`q`-quantile of a*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)`

.

__Function:__**mean_weibull***(*`a`,`b`)Returns the mean of a

*Weibull(a,b)*random variable, with*a,b>0*. To make use of this function, write first`load(distrib)`

.

__Function:__**var_weibull***(*`a`,`b`)Returns the variance of a

*Weibull(a,b)*random variable, with*a,b>0*. To make use of this function, write first`load(distrib)`

.

__Function:__**std_weibull***(*`a`,`b`)Returns the standard deviation of a

*Weibull(a,b)*random variable, with*a,b>0*. To make use of this function, write first`load(distrib)`

.

__Function:__**skewness_weibull***(*`a`,`b`)Returns the skewness coefficient of a

*Weibull(a,b)*random variable, with*a,b>0*. To make use of this function, write first`load(distrib)`

.

__Function:__**kurtosis_weibull***(*`a`,`b`)Returns the kurtosis coefficient of a

*Weibull(a,b)*random variable, with*a,b>0*. To make use of this function, write first`load(distrib)`

.

__Function:__**random_weibull***(*`a`,`b`)__Function:__**random_weibull***(*`a`,`b`,`n`)Returns a

*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)`

.

__Function:__**pdf_rayleigh***(*`x`,`b`)Returns the value at

`x`of the density function of a*Rayleigh(b)*random variable, with*b>0*.The

*Rayleigh(b)*random variable is equivalent to the*Weibull(2,1/b)*, therefore when Maxima has not enough information to get the result, a noun form based on the Weibull density is returned.(%i1) load (distrib)$ (%i2) pdf_rayleigh(x,b); 1 (%o2) pdf_weibull(x, 2, -) b (%i3) assume(x>0,b>0)$ pdf_rayleigh(x,b); 2 2 2 - b x (%o4) 2 b x %e

__Function:__**cdf_rayleigh***(*`x`,`b`)Returns the value at

`x`of the distribution function of a*Rayleigh(b)*random variable, with*b>0*.The

*Rayleigh(b)*random variable is equivalent to the*Weibull(2,1/b)*, therefore when Maxima has not enough information to get the result, a noun form based on the Weibull distribution is returned.(%i1) load (distrib)$ (%i2) cdf_rayleigh(x,b); 1 (%o2) cdf_weibull(x, 2, -) b (%i3) assume(x>0,b>0)$ cdf_rayleigh(x,b); 2 2 - b x (%o4) 1 - %e

__Function:__**quantile_rayleigh***(*`q`,`b`)Returns the

`q`-quantile of a*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

*Rayleigh(b)*random variable is equivalent to the*Weibull(2,1/b)*, therefore when Maxima has not enough information to get the result, a noun form based on the Weibull quantile is returned.(%i1) load (distrib)$ (%i2) quantile_rayleigh(0.99,b); 1 (%o2) quantile_weibull(0.99, 2, -) b (%i3) assume(x>0,b>0)$ quantile_rayleigh(0.99,b); 2.145966026289347 (%o4) ----------------- b

__Function:__**mean_rayleigh***(*`b`)Returns the mean of a

*Rayleigh(b)*random variable, with*b>0*.The

*Rayleigh(b)*random variable is equivalent to the*Weibull(2,1/b)*, therefore when Maxima has not enough information to get the result, a noun form based on the Weibull mean is returned.(%i1) load (distrib)$ (%i2) mean_rayleigh(b); 1 (%o2) mean_weibull(2, -) b (%i3) assume(b>0)$ mean_rayleigh(b); sqrt(%pi) (%o4) --------- 2 b

__Function:__**var_rayleigh***(*`b`)Returns the variance of a

*Rayleigh(b)*random variable, with*b>0*.The

*Rayleigh(b)*random variable is equivalent to the*Weibull(2,1/b)*, therefore when Maxima has not enough information to get the result, a noun form based on the Weibull variance is returned.(%i1) load (distrib)$ (%i2) var_rayleigh(b); 1 (%o2) var_weibull(2, -) b (%i3) assume(b>0)$ var_rayleigh(b); %pi 1 - --- 4 (%o4) ------- 2 b

__Function:__**std_rayleigh***(*`b`)Returns the standard deviation of a

*Rayleigh(b)*random variable, with*b>0*.The

*Rayleigh(b)*random variable is equivalent to the*Weibull(2,1/b)*, therefore when Maxima has not enough information to get the result, a noun form based on the Weibull standard deviation is returned.(%i1) load (distrib)$ (%i2) std_rayleigh(b); 1 (%o2) std_weibull(2, -) b (%i3) assume(b>0)$ std_rayleigh(b); %pi sqrt(1 - ---) 4 (%o4) ------------- b

__Function:__**skewness_rayleigh***(*`b`)Returns the skewness coefficient of a

*Rayleigh(b)*random variable, with*b>0*.The

*Rayleigh(b)*random variable is equivalent to the*Weibull(2,1/b)*, therefore when Maxima has not enough information to get the result, a noun form based on the Weibull skewness coefficient is returned.(%i1) load (distrib)$ (%i2) skewness_rayleigh(b); 1 (%o2) skewness_weibull(2, -) b (%i3) assume(b>0)$ skewness_rayleigh(b); 3/2 %pi 3 sqrt(%pi) ------ - ----------- 4 4 (%o4) -------------------- %pi 3/2 (1 - ---) 4

__Function:__**kurtosis_rayleigh***(*`b`)Returns the kurtosis coefficient of a

*Rayleigh(b)*random variable, with*b>0*.The

*Rayleigh(b)*random variable is equivalent to the*Weibull(2,1/b)*, therefore when Maxima has not enough information to get the result, a noun form based on the Weibull kurtosis coefficient is returned.(%i1) load (distrib)$ (%i2) kurtosis_rayleigh(b); 1 (%o2) kurtosis_weibull(2, -) b (%i3) assume(b>0)$ kurtosis_rayleigh(b); 2 3 %pi 2 - ------ 16 (%o4) ---------- - 3 %pi 2 (1 - ---) 4

__Function:__**random_rayleigh***(*`b`)__Function:__**random_rayleigh***(*`b`,`n`)Returns a

*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)`

.

__Function:__**pdf_laplace***(*`x`,`a`,`b`)Returns the value at

`x`of the density function of a*Laplace(a,b)*random variable, with*b>0*. To make use of this function, write first`load(distrib)`

.

__Function:__**cdf_laplace***(*`x`,`a`,`b`)Returns the value at

`x`of the distribution function of a*Laplace(a,b)*random variable, with*b>0*. To make use of this function, write first`load(distrib)`

.

__Function:__**quantile_laplace***(*`q`,`a`,`b`)Returns the

`q`-quantile of a*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)`

.

__Function:__**mean_laplace***(*`a`,`b`)Returns the mean of a

*Laplace(a,b)*random variable, with*b>0*. To make use of this function, write first`load(distrib)`

.

__Function:__**var_laplace***(*`a`,`b`)Returns the variance of a

*Laplace(a,b)*random variable, with*b>0*. To make use of this function, write first`load(distrib)`

.

__Function:__**std_laplace***(*`a`,`b`)Returns the standard deviation of a

*Laplace(a,b)*random variable, with*b>0*. To make use of this function, write first`load(distrib)`

.

__Function:__**skewness_laplace***(*`a`,`b`)Returns the skewness coefficient of a

*Laplace(a,b)*random variable, with*b>0*. To make use of this function, write first`load(distrib)`

.

__Function:__**kurtosis_laplace***(*`a`,`b`)Returns the kurtosis coefficient of a

*Laplace(a,b)*random variable, with*b>0*. To make use of this function, write first`load(distrib)`

.

__Function:__**random_laplace***(*`a`,`b`)__Function:__**random_laplace***(*`a`,`b`,`n`)Returns a

*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)`

.

__Function:__**pdf_cauchy***(*`x`,`a`,`b`)Returns the value at

`x`of the density function of a*Cauchy(a,b)*random variable, with*b>0*. To make use of this function, write first`load(distrib)`

.

__Function:__**cdf_cauchy***(*`x`,`a`,`b`)Returns the value at

`x`of the distribution function of a*Cauchy(a,b)*random variable, with*b>0*. To make use of this function, write first`load(distrib)`

.

__Function:__**quantile_cauchy***(*`q`,`a`,`b`)Returns the

`q`-quantile of a*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)`

.

__Function:__**random_cauchy***(*`a`,`b`)__Function:__**random_cauchy***(*`a`,`b`,`n`)Returns a

*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)`

.

__Function:__**pdf_gumbel***(*`x`,`a`,`b`)Returns the value at

`x`of the density function of a*Gumbel(a,b)*random variable, with*b>0*. To make use of this function, write first`load(distrib)`

.

__Function:__**cdf_gumbel***(*`x`,`a`,`b`)Returns the value at

`x`of the distribution function of a*Gumbel(a,b)*random variable, with*b>0*. To make use of this function, write first`load(distrib)`

.

__Function:__**quantile_gumbel***(*`q`,`a`,`b`)Returns the

`q`-quantile of a*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)`

.

__Function:__**mean_gumbel***(*`a`,`b`)Returns the mean of a

*Gumbel(a,b)*random variable, with*b>0*.(%i1) load (distrib)$ (%i2) assume(b>0)$ mean_gumbel(a,b); (%o3) %gamma b + a

where symbol

`%gamma`

stands for the Euler-Mascheroni constant. See also`%gamma`

.

__Function:__**var_gumbel***(*`a`,`b`)Returns the variance of a

*Gumbel(a,b)*random variable, with*b>0*. To make use of this function, write first`load(distrib)`

.

__Function:__**std_gumbel***(*`a`,`b`)Returns the standard deviation of a

*Gumbel(a,b)*random variable, with*b>0*. To make use of this function, write first`load(distrib)`

.

__Function:__**skewness_gumbel***(*`a`,`b`)Returns the skewness coefficient of a

*Gumbel(a,b)*random variable, with*b>0*.(%i1) load (distrib)$ (%i2) assume(b>0)$ skewness_gumbel(a,b); 12 sqrt(6) zeta(3) (%o3) ------------------ 3 %pi (%i4) numer:true$ skewness_gumbel(a,b); (%o5) 1.139547099404649

where

`zeta`

stands for the Riemann's zeta function.

__Function:__**kurtosis_gumbel***(*`a`,`b`)Returns the kurtosis coefficient of a

*Gumbel(a,b)*random variable, with*b>0*. To make use of this function, write first`load(distrib)`

.

__Function:__**random_gumbel***(*`a`,`b`)__Function:__**random_gumbel***(*`a`,`b`,`n`)Returns a

*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)`

.

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__Function:__**pdf_general_finite_discrete***(*`x`,`v`)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)`

.Examples:

(%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

__Function:__**cdf_general_finite_discrete***(*`x`,`v`)Returns the value at

`x`of the distribution function of a general finite discrete random variable, with vector probabilities*v*.See

`pdf_general_finite_discrete`

for more details.Examples:

(%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

__Function:__**quantile_general_finite_discrete***(*`q`,`v`)Returns the

`q`-quantile of a general finite discrete random variable, with vector probabilities*v*.See

`pdf_general_finite_discrete`

for more details.

__Function:__**mean_general_finite_discrete***(*`v`)Returns the mean of a general finite discrete random variable, with vector probabilities

*v*.See

`pdf_general_finite_discrete`

for more details.

__Function:__**var_general_finite_discrete***(*`v`)Returns the variance of a general finite discrete random variable, with vector probabilities

*v*.See

`pdf_general_finite_discrete`

for more details.

__Function:__**std_general_finite_discrete***(*`v`)Returns the standard deviation of a general finite discrete random variable, with vector probabilities

*v*.See

`pdf_general_finite_discrete`

for more details.

__Function:__**skewness_general_finite_discrete***(*`v`)Returns the skewness coefficient of a general finite discrete random variable, with vector probabilities

*v*.See

`pdf_general_finite_discrete`

for more details.

__Function:__**kurtosis_general_finite_discrete***(*`v`)Returns the kurtosis coefficient of a general finite discrete random variable, with vector probabilities

*v*.See

`pdf_general_finite_discrete`

for more details.

__Function:__**random_general_finite_discrete***(*`v`)__Function:__**random_general_finite_discrete***(*`v`,`m`)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_general_finite_discrete`

for more details.Examples:

(%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, 2, 2, 3, 2, 4, 4, 1, 2, 2]

__Function:__**pdf_binomial***(*`x`,`n`,`p`)Returns the value at

`x`of the probability function of a*Binomial(n,p)*random variable, with*0<p<1*and*n*a positive integer. To make use of this function, write first`load(distrib)`

.

__Function:__**cdf_binomial***(*`x`,`n`,`p`)Returns the value at

`x`of the distribution function of a*Binomial(n,p)*random variable, with*0<p<1*and*n*a positive integer.(%i1) load (distrib)$ (%i2) cdf_binomial(5,7,1/6); 7775 (%o2) ---- 7776 (%i3) float(%); (%o3) .9998713991769548

__Function:__**quantile_binomial***(*`q`,`n`,`p`)Returns the

`q`-quantile of a*Binomial(n,p)*random variable, with*0<p<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)`

.

__Function:__**mean_binomial***(*`n`,`p`)Returns the mean of a

*Binomial(n,p)*random variable, with*0<p<1*and*n*a positive integer. To make use of this function, write first`load(distrib)`

.

__Function:__**var_binomial***(*`n`,`p`)Returns the variance of a

*Binomial(n,p)*random variable, with*0<p<1*and*n*a positive integer. To make use of this function, write first`load(distrib)`

.

__Function:__**std_binomial***(*`n`,`p`)Returns the standard deviation of a

*Binomial(n,p)*random variable, with*0<p<1*and*n*a positive integer. To make use of this function, write first`load(distrib)`

.

__Function:__**skewness_binomial***(*`n`,`p`)Returns the skewness coefficient of a

*Binomial(n,p)*random variable, with*0<p<1*and*n*a positive integer. To make use of this function, write first`load(distrib)`

.

__Function:__**kurtosis_binomial***(*`n`,`p`)Returns the kurtosis coefficient of a

*Binomial(n,p)*random variable, with*0<p<1*and*n*a positive integer. To make use of this function, write first`load(distrib)`

.

__Function:__**random_binomial***(*`n`,`p`)__Function:__**random_binomial***(*`n`,`p`,`m`)Returns a

*Binomial(n,p)*random variate, with*0<p<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)`

.

__Function:__**pdf_poisson***(*`x`,`m`)Returns the value at

`x`of the probability function of a*Poisson(m)*random variable, with*m>0*. To make use of this function, write first`load(distrib)`

.

__Function:__**cdf_poisson***(*`x`,`m`)Returns the value at

`x`of the distribution function of a*Poisson(m)*random variable, with*m>0*.(%i1) load (distrib)$ (%i2) cdf_poisson(3,5); (%o2) gamma_incomplete_regularized(4, 5) (%i3) float(%); (%o3) .2650259152973623

__Function:__**quantile_poisson***(*`q`,`m`)Returns the

`q`-quantile of a*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)`

.

__Function:__**mean_poisson***(*`m`)Returns the mean of a

*Poisson(m)*random variable, with*m>0*. To make use of this function, write first`load(distrib)`

.

__Function:__**var_poisson***(*`m`)Returns the variance of a

*Poisson(m)*random variable, with*m>0*. To make use of this function, write first`load(distrib)`

.

__Function:__**std_poisson***(*`m`)Returns the standard deviation of a

*Poisson(m)*random variable, with*m>0*. To make use of this function, write first`load(distrib)`

.

__Function:__**skewness_poisson***(*`m`)Returns the skewness coefficient of a

*Poisson(m)*random variable, with*m>0*. To make use of this function, write first`load(distrib)`

.

__Function:__**kurtosis_poisson***(*`m`)Returns the kurtosis coefficient of a Poisson random variable

*Poi(m)*, with*m>0*. To make use of this function, write first`load(distrib)`

.

__Function:__**random_poisson***(*`m`)__Function:__**random_poisson***(*`m`,`n`)Returns a

*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)`

.

__Function:__**pdf_bernoulli***(*`x`,`p`)Returns the value at

`x`of the probability function of a*Bernoulli(p)*random variable, with*0<p<1*.The

*Bernoulli(p)*random variable is equivalent to the*Binomial(1,p)*, therefore when Maxima has not enough information to get the result, a noun form based on the binomial probability function is returned.(%i1) load (distrib)$ (%i2) pdf_bernoulli(1,p); (%o2) pdf_binomial(1, 1, p) (%i3) assume(0<p,p<1)$ pdf_bernoulli(1,p); (%o4) p

__Function:__**cdf_bernoulli***(*`x`,`p`)Returns the value at

`x`of the distribution function of a*Bernoulli(p)*random variable, with*0<p<1*. To make use of this function, write first`load(distrib)`

.

__Function:__**quantile_bernoulli***(*`q`,`p`)Returns the

`q`-quantile of a*Bernoulli(p)*random variable, with*0<p<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)`

.

__Function:__**mean_bernoulli***(*`p`)Returns the mean of a

*Bernoulli(p)*random variable, with*0<p<1*.The

*Bernoulli(p)*random variable is equivalent to the*Binomial(1,p)*, therefore when Maxima has not enough information to get the result, a noun form based on the binomial mean is returned.(%i1) load (distrib)$ (%i2) mean_bernoulli(p); (%o2) mean_binomial(1, p) (%i3) assume(0<p,p<1)$ mean_bernoulli(p); (%o4) p

__Function:__**var_bernoulli***(*`p`)Returns the variance of a

*Bernoulli(p)*random variable, with*0<p<1*.The

*Bernoulli(p)*random variable is equivalent to the*Binomial(1,p)*, therefore when Maxima has not enough information to get the result, a noun form based on the binomial variance is returned.(%i1) load (distrib)$ (%i2) var_bernoulli(p); (%o2) var_binomial(1, p) (%i3) assume(0<p,p<1)$ var_bernoulli(p); (%o4) (1 - p) p

__Function:__**std_bernoulli***(*`p`)Returns the standard deviation of a

*Bernoulli(p)*random variable, with*0<p<1*.The

*Bernoulli(p)*random variable is equivalent to the*Binomial(1,p)*, therefore when Maxima has not enough information to get the result, a noun form based on the binomial standard deviation is returned.(%i1) load (distrib)$ (%i2) std_bernoulli(p); (%o2) std_binomial(1, p) (%i3) assume(0<p,p<1)$ std_bernoulli(p); (%o4) sqrt(1 - p) sqrt(p)

__Function:__**skewness_bernoulli***(*`p`)Returns the skewness coefficient of a

*Bernoulli(p)*random variable, with*0<p<1*.The

*Bernoulli(p)*random variable is equivalent to the*Binomial(1,p)*, therefore when Maxima has not enough information to get the result, a noun form based on the binomial skewness coefficient is returned.(%i1) load (distrib)$ (%i2) skewness_bernoulli(p); (%o2) skewness_binomial(1, p) (%i3) assume(0<p,p<1)$ skewness_bernoulli(p); 1 - 2 p (%o4) ------------------- sqrt(1 - p) sqrt(p)

__Function:__**kurtosis_bernoulli***(*`p`)Returns the kurtosis coefficient of a

*Bernoulli(p)*random variable, with*0<p<1*.The

*Bernoulli(p)*random variable is equivalent to the*Binomial(1,p)*, therefore when Maxima has not enough information to get the result, a noun form based on the binomial kurtosis coefficient is returned.(%i1) load (distrib)$ (%i2) kurtosis_bernoulli(p); (%o2) kurtosis_binomial(1, p) (%i3) assume(0<p,p<1)$ kurtosis_bernoulli(p); 1 - 6 (1 - p) p (%o4) --------------- (1 - p) p

__Function:__**random_bernoulli***(*`p`)__Function:__**random_bernoulli***(*`p`,`n`)Returns a

*Bernoulli(p)*random variate, with*0<p<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)`

.

__Function:__**pdf_geometric***(*`x`,`p`)Returns the value at

`x`of the probability function of a*Geometric(p)*random variable, with*0<p<1*. To make use of this function, write first`load(distrib)`

.

__Function:__**cdf_geometric***(*`x`,`p`)Returns the value at

`x`of the distribution function of a*Geometric(p)*random variable, with*0<p<1*. To make use of this function, write first`load(distrib)`

.

__Function:__**quantile_geometric***(*`q`,`p`)Returns the

`q`-quantile of a*Geometric(p)*random variable, with*0<p<1*; in other words, this is the inverse of`cdf_geometric`

. Argument`q`must be an element of*[0,1]*. To make use of this function, write first`load(distrib)`

.

__Function:__**mean_geometric***(*`p`)Returns the mean of a

*Geometric(p)*random variable, with*0<p<1*. To make use of this function, write first`load(distrib)`

.

__Function:__**var_geometric***(*`p`)Returns the variance of a

*Geometric(p)*random variable, with*0<p<1*. To make use of this function, write first`load(distrib)`

.

__Function:__**std_geometric***(*`p`)Returns the standard deviation of a

*Geometric(p)*random variable, with*0<p<1*. To make use of this function, write first`load(distrib)`

.

__Function:__**skewness_geometric***(*`p`)Returns the skewness coefficient of a

*Geometric(p)*random variable, with*0<p<1*. To make use of this function, write first`load(distrib)`

.

__Function:__**kurtosis_geometric***(*`p`)Returns the kurtosis coefficient of a geometric random variable

*Geo(p)*, with*0<p<1*. To make use of this function, write first`load(distrib)`

.

__Function:__**random_geometric***(*`p`)__Function:__**random_geometric***(*`p`,`n`)Returns a

*Geometric(p)*random variate, with*0<p<1*. Calling`random_geometric`

with a second argument`n`, a random sample of size`n`will be simulated.The algorithm is based on simulation of Bernoulli trials.

To make use of this function, write first

`load(distrib)`

.

__Function:__**pdf_discrete_uniform***(*`x`,`n`)Returns the value at

`x`of the probability function of a*Discrete Uniform(n)*random variable, with*n*a strictly positive integer. To make use of this function, write first`load(distrib)`

.

__Function:__**cdf_discrete_uniform***(*`x`,`n`)Returns the value at

`x`of the distribution function of a*Discrete Uniform(n)*random variable, with*n*a strictly positive integer. To make use of this function, write first`load(distrib)`

.

__Function:__**quantile_discrete_uniform***(*`q`,`n`)Returns the

`q`-quantile of a*Discrete Uniform(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)`

.

__Function:__**mean_discrete_uniform***(*`n`)Returns the mean of a

*Discrete Uniform(n)*random variable, with*n*a strictly positive integer. To make use of this function, write first`load(distrib)`

.

__Function:__**var_discrete_uniform***(*`n`)Returns the variance of a

*Discrete Uniform(n)*random variable, with*n*a strictly positive integer. To make use of this function, write first`load(distrib)`

.

__Function:__**std_discrete_uniform***(*`n`)Returns the standard deviation of a

*Discrete Uniform(n)*random variable, with*n*a strictly positive integer. To make use of this function, write first`load(distrib)`

.

__Function:__**skewness_discrete_uniform***(*`n`)Returns the skewness coefficient of a

*Discrete Uniform(n)*random variable, with*n*a strictly positive integer. To make use of this function, write first`load(distrib)`

.

__Function:__**kurtosis_discrete_uniform***(*`n`)Returns the kurtosis coefficient of a

*Discrete Uniform(n)*random variable, with*n*a strictly positive integer. To make use of this function, write first`load(distrib)`

.

__Function:__**random_discrete_uniform***(*`n`)__Function:__**random_discrete_uniform***(*`n`,`m`)Returns a

*Discrete Uniform(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)`

.

__Function:__**pdf_hypergeometric***(*`x`,`n1`,`n2`,`n`)Returns the value at

`x`of the probability function of a*Hypergeometric(n1,n2,n)*random variable, with`n1`,`n2`and`n`non negative integers and*n<=n1+n2*. Being`n1`the number of objects of class A,`n2`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)`

.

__Function:__**cdf_hypergeometric***(*`x`,`n1`,`n2`,`n`)Returns the value at

`x`of the distribution function of a*Hypergeometric(n1,n2,n)*random variable, with`n1`,`n2`and`n`non negative integers and*n<=n1+n2*. See`pdf_hypergeometric`

for a more complete description.To make use of this function, write first

`load(distrib)`

.

__Function:__**quantile_hypergeometric***(*`q`,`n1`,`n2`,`n`)Returns the

`q`-quantile of a*Hypergeometric(n1,n2,n)*random variable, with`n1`,`n2`and`n`non negative integers and*n<=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)`

.

__Function:__**mean_hypergeometric***(*`n1`,`n2`,`n`)Returns the mean of a discrete uniform random variable

*Hyp(n1,n2,n)*, with`n1`,`n2`and`n`non negative integers and*n<=n1+n2*. To make use of this function, write first`load(distrib)`

.

__Function:__**var_hypergeometric***(*`n1`,`n2`,`n`)Returns the variance of a hypergeometric random variable

*Hyp(n1,n2,n)*, with`n1`,`n2`and`n`non negative integers and*n<=n1+n2*. To make use of this function, write first`load(distrib)`

.

__Function:__**std_hypergeometric***(*`n1`,`n2`,`n`)Returns the standard deviation of a

*Hypergeometric(n1,n2,n)*random variable, with`n1`,`n2`and`n`non negative integers and*n<=n1+n2*. To make use of this function, write first`load(distrib)`

.

__Function:__**skewness_hypergeometric***(*`n1`,`n2`,`n`)Returns the skewness coefficient of a

*Hypergeometric(n1,n2,n)*random variable, with`n1`,`n2`and`n`non negative integers and*n<=n1+n2*. To make use of this function, write first`load(distrib)`

.

__Function:__**kurtosis_hypergeometric***(*`n1`,`n2`,`n`)Returns the kurtosis coefficient of a

*Hypergeometric(n1,n2,n)*random variable, with`n1`,`n2`and`n`non negative integers and*n<=n1+n2*. To make use of this function, write first`load(distrib)`

.

__Function:__**random_hypergeometric***(*`n1`,`n2`,`n`)__Function:__**random_hypergeometric***(*`n1`,`n2`,`n`,`m`)Returns a

*Hypergeometric(n1,n2,n)*random variate, with`n1`,`n2`and`n`non negative integers and*n<=n1+n2*. 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)`

.

__Function:__**pdf_negative_binomial***(*`x`,`n`,`p`)Returns the value at

`x`of the probability function of a*Negative Binomial(n,p)*random variable, with*0<p<1*and*n*a positive integer. To make use of this function, write first`load(distrib)`

.

__Function:__**cdf_negative_binomial***(*`x`,`n`,`p`)Returns the value at

`x`of the distribution function of a*Negative Binomial(n,p)*random variable, with*0<p<1*and*n*a positive integer.(%i1) load (distrib)$ (%i2) cdf_negative_binomial(3,4,1/8); 3271 (%o2) ------ 524288 (%i3) float(%); (%o3) .006238937377929687

__Function:__**quantile_negative_binomial***(*`q`,`n`,`p`)Returns the

`q`-quantile of a*Negative Binomial(n,p)*random variable, with*0<p<1*and*n*a positive integer; 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)`

.

__Function:__**mean_negative_binomial***(*`n`,`p`)Returns the mean of a

*Negative Binomial(n,p)*random variable, with*0<p<1*and*n*a positive integer. To make use of this function, write first`load(distrib)`

.

__Function:__**var_negative_binomial***(*`n`,`p`)Returns the variance of a

*Negative Binomial(n,p)*random variable, with*0<p<1*and*n*a positive integer. To make use of this function, write first`load(distrib)`

.

__Function:__**std_negative_binomial***(*`n`,`p`)Returns the standard deviation of a

*Negative Binomial(n,p)*random variable, with*0<p<1*and*n*a positive integer. To make use of this function, write first`load(distrib)`

.

__Function:__**skewness_negative_binomial***(*`n`,`p`)Returns the skewness coefficient of a

*Negative Binomial(n,p)*random variable, with*0<p<1*and*n*a positive integer. To make use of this function, write first`load(distrib)`

.

__Function:__**kurtosis_negative_binomial***(*`n`,`p`)Returns the kurtosis coefficient of a

*Negative Binomial(n,p)*random variable, with*0<p<1*and*n*a positive integer. To make use of this function, write first`load(distrib)`

.

__Function:__**random_negative_binomial***(*`n`,`p`)__Function:__**random_negative_binomial***(*`n`,`p`,`m`)Returns a

*Negative Binomial(n,p)*random variate, with*0<p<1*and*n*a positive integer. 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)`

.

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