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 cumulative distribution function is defined as
\[F\left(x\right)=\int_{ -\infty }^{x}{f\left(u\right)\;du}
\]
which equals the probability \({\rm Pr}(X \le x).\)
The mean value is a localization parameter and is defined as
\[E\left[X\right]=\int_{ -\infty }^{\infty }{x\,f\left(x\right)\;dx}
\]
The variance is a measure of variation,
\[V\left[X\right]=\int_{ -\infty }^{\infty }{f\left(x\right)\,\left(x
-E\left[X\right]\right)^2\;dx}
\]
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,
\[SK\left[X\right]={{\int_{ -\infty }^{\infty }{f\left(x\right)\,
\left(x-E\left[X\right]\right)^3\;dx}}\over{D\left[X\right]^3}}
\]
And the kurtosis coefficient measures the peakedness of the distribution,
\[KU\left[X\right]={{\int_{ -\infty }^{\infty }{f\left(x\right)\,
\left(x-E\left[X\right]\right)^4\;dx}}\over{D\left[X\right]^4}}-3
\]
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 cumulative distribution function is
\[ F\left(x\right)=\sum_{x_{i}\leq x}{f\left(x_{i}\right)}
\]
The mean, variance, standard deviation, skewness coefficient and kurtosis coefficient take the form
\[\eqalign{
E\left[X\right]&=\sum_{x_{i}}{x_{i}f\left(x_{i}\right)}, \cr
V\left[X\right]&=\sum_{x_{i}}{f\left(x_{i}\right)\left(x_{i}-E\left[X\right]\right)^2},\cr
D\left[X\right]&=\sqrt{V\left[X\right]},\cr
SK\left[X\right]&={{\sum_{x_{i}}{f\left(x\right)\,
\left(x-E\left[X\right]\right)^3\;dx}}\over{D\left[X\right]^3}}, \cr
KU\left[X\right]&={{\sum_{x_{i}}{f\left(x\right)\,
\left(x-E\left[X\right]\right)^4\;dx}}\over{D\left[X\right]^4}}-3,
}
\]
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 ’riotorto AT yahoo DOT com’.
Maxima knows the following kinds of continuous distributions.
Normal random variables (also called Gaussian) is denoted by \({\it Normal}(m, s)\) where \(m\) is the mean and \(s > 0\) is the standard deviation.
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
\[f(x; m, s) = {1\over s\sqrt{2\pi}} e^{\displaystyle -{(x-m)^2\over 2s^2}}
\]
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:
\[F(x; m, s) = {1\over 2} + {1\over 2} {\rm erf}\left(x-m\over s\sqrt{2}\right)
\]
(%i1) load ("distrib")$
(%i2) cdf_normal(x,m,s);
x - m
erf(---------)
sqrt(2) s 1
(%o2) -------------- + -
2 2
See also erf.
Returns the q-quantile of a
\({\it Normal}(m, s)\)
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
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
\[E[X] = m \]
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
\[V[X] = s^2 \]
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
\[D[X] = s \]
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
\[SK[X] = 0 \]
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
\[KU[X] = 0 \]
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").
Student’s t random variable is denoted by \(t(n)\) where \(n\) is the degrees of freedom with \(n > 0\). If \(Z\) is a \({\it Normal}(0, 1)\) variable and \(V\) is an independent \(\chi^2\) random variable with \(n\) degress of freedom, then
\[Z \over \sqrt{V/n}
\]
has a Student’s \(t\)-distribution with \(n\) degrees of freedom.
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
\[f(x; n) = \left[\sqrt{n} B\left({1\over 2}, {n\over 2}\right)\right]^{-1}
\left(1+{x^2\over n}\right)^{\displaystyle -{n+1\over 2}}
\]
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
\[F(x; n) =
\cases{
1-\displaystyle{1\over 2} I_t\left({n\over 2}, {1\over 2}\right) & $x \ge 0$ \cr
\cr
\displaystyle{1\over 2} I_t\left({n\over 2}, {1\over 2}\right) & $x < 0$
}
\]
where \(t = n/(n+x^2)\) and \(I_t(a,b)\) is the beta_incomplete_regularized 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
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").
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
\[E[X] = 0 \]
Returns the variance of a Student random variable \(t(n)\) , with \(n>2\).
The variance is
\[V[X] = {n\over n-2}
\]
(%i1) load ("distrib")$
(%i2) var_student_t(n);
n
(%o2) -----
n - 2
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
\[D[X] = \sqrt{\displaystyle{n\over n-2}}
\]
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
\[SK[X] = 0 \]
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
\[KU[X] = {6\over n-4}
\]
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
\[X={{Z}\over{\sqrt{{S^2}\over{n}}}}
\]
is a Student random variable with \(n\) degrees of freedom, \(t(n)\) .
To make use of this function, write first load("distrib").
Let \(ncp\) be the non-centrality parameter, \(n\) be the degrees of freedom for the non-central Student’s \(t\) random variable.
Then let \(X\) be a \({\it Normal}(n, ncp)\) and \(S^2\) be an independent \(\chi^2\) random variable with \(n\) degrees of freedom, the random variable
\[U = {X \over \sqrt{S^2\over n}}
\]
has a non-central Student’s \(t\) distribution with non-centrality parameter \(ncp\).
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
\[f(x; n, \mu) = \left[\sqrt{n} B\left({1\over 2}, {n\over
2}\right)\right]^{-1}\left(1+{x^2\over n}\right)^{-{(n+1)/2}}
e^{-\mu^2/ 2}
\bigg[A_n(x; \mu) + B_n(x; \mu)\bigg]
\]
where
\[\eqalign{
A_n(x;\mu) &= {}_1F_1\left({n+1\over 2}; {1\over 2}; {\mu^2 x^2\over
2\left(x^2+n\right)}\right) \cr
B_n(x;\mu) &= {\sqrt{2}\mu x \over \sqrt{x^2+n}} {\Gamma\left({n\over
2} + 1\right)\over \Gamma\left({n+1\over 2}\right)}\;
{}_1F_1\left({n\over 2} + 1; {3\over 2}; {\mu^2 x^2\over
2\left(x^2+n\right)}\right)
}
\]
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 0.01889822365046136 by 15934951/843198350 = 0.01889822365046136
rat: replaced -8.734356480209641 by -294697965/33740089 = -8.734356480209641
rat: replaced 4.136255165816327 by 51033443/12338079 = 4.136255165816332
rat: replaced 1.08061432164203 by 56754827/52520891 = 1.08061432164203
rat: replaced 0.0565127306411839 by 5608717/99246965 = 0.05651273064118384
rat: replaced -300.8069396896258 by -79782423/265228 = -300.8069396896256
rat: replaced 160.6269176184973 by 178374907/1110492 = 160.626917618497
7/2 7/2
0.04296414417400905 5 1.323650307289301e-6 5
(%o2) ------------------------ + -------------------------
3/2 5/2 sqrt(%pi)
2 14 sqrt(%pi)
7/2
1.94793720435093e-4 5
+ ------------------------
%pi
(%i3) float(%); (%o3) 0.02080593159405671
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
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").
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
\[E[X] = {\mu \sqrt{n}\; \Gamma\left(\displaystyle{n-1\over 2}\right) \over
\sqrt{2}\;\Gamma\left(\displaystyle{n\over 2}\right)}
\]
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
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
\[V[X] = {n(\mu^2+1)\over n-2} - {n\mu^2\; \Gamma\left(\displaystyle{n-1\over 2}\right)^2
\over 2\Gamma\left(\displaystyle{n\over 2}\right)^2}
\]
where \(\mu\) is the noncentrality parameter \(ncp\).
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
\[D[X] = \sqrt{{n(\mu^2+1)\over n-2} - {n\mu^2\; \Gamma\left(\displaystyle{n-1\over 2}\right)^2
\over 2\Gamma\left(\displaystyle{n\over 2}\right)^2}}
\]
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
\[\eqalign{
SK[U] &=
{\mu\sqrt{n}\,\Gamma\left({{n-1}\over{2}}\right)
\over{\sqrt{2}\Gamma\left({{n
}\over{2}}\right)\sigma^{3}}}\left({{n
\left(2n+\mu^2-3\right)}\over{\left(n-3\right)\left(n-2\right)}}
-2\sigma^2\right) \cr
\sigma^2 &= {{n\left(\mu^2+1\right)}\over{n-2}}-{{n \mu^2\,
\Gamma\left({{n-1}\over{2}}\right)^2}\over{2\Gamma\left({{n
}\over{2}}\right)^2}}
}
\]
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
\[\eqalign{
KU[U] &=
{\mu_4\over \sigma^4} - 3\cr
\mu_4 &= {{\left(\mu^4+6\mu^2+3\right)n^2}\over{(n-4)(n-2)}}
-\left({{n\left(3(3n-5)+\mu^2(n+1)\right)
}\over{(n-3)(n-2)}}-3\sigma^2\right) F \cr
\sigma^2 &= {{n\left(\mu^2+1\right)}\over{n-2}}-{{n \mu^2
\Gamma\left({{n-1}\over{2}}\right)^2}\over{2\Gamma\left({{n
}\over{2}}\right)^2}} \cr
F &= {n\mu^2\Gamma\left({n-1\over 2}\right)^2 \over
2\sigma^4\Gamma\left({n\over 2}\right)^2}
}
\]
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
\[U={{X}\over{\sqrt{{S^2}\over{n}}}}
\]
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").
Let \(X_1, X_2, \ldots, X_n\) be independent and identically distributed \({\it Normal}(0, 1)\) variables. Then
\[X^2 = \sum_{i=1}^n X_i^2
\]
is said to follow a chi-square distribution with \(n\) degrees of freedom.
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
\[f(x; n) =
\cases{
\displaystyle{x^{n/2-1} e^{-x/2} \over 2^{n/2}
\Gamma\left(\displaystyle{n\over 2}\right)} & for $x
> 0$ \cr
\cr
0 & otherwise
}
\]
(%i1) load ("distrib")$
(%i2) pdf_chi2(x,n);
n/2 - 1 - x/2
x %e unit_step(x)
(%o2) -----------------------------
n n/2
gamma(-) 2
2
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
\[F(x; n) =
\cases{
1 - Q\left(\displaystyle{n\over 2}, {x\over 2}\right) & $x > 0$ \cr
0 & otherwise
}
\]
where \(Q(a,z)\) is the gamma_incomplete_regularized function.
(%i1) load ("distrib")$
(%i2) cdf_chi2(3,4);
3
(%o2) 1 - gamma_incomplete_regularized(2, -)
2
(%i3) float(%); (%o3) 0.4421745996289252
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
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
\[E[X] = n \]
(%i1) load ("distrib")$
(%i2) mean_chi2(n); (%o2) 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 \(\Gamma\left(n/2,2\right)\) .
The variance is
\[V[X] = 2n \]
(%i1) load ("distrib")$
(%i2) var_chi2(n); (%o2) 2 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 \(\Gamma\left(n/2,2\right)\) .
The standard deviation is
\[D[X] = \sqrt{2n}
\]
(%i1) load ("distrib")$
(%i2) std_chi2(n); (%o2) sqrt(2) sqrt(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 \(\Gamma\left(n/2,2\right)\) .
The skewness coefficient is
\[SK[X] = \sqrt{8\over n}
\]
(%i1) load ("distrib")$
(%i2) skewness_chi2(n);
3/2
2
(%o2) -------
sqrt(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 \(\Gamma\left(n/2,2\right)\) .
The kurtosis coefficient is
\[KU[X] = {12\over n}
\]
(%i1) load ("distrib")$
(%i2) kurtosis_chi2(n);
12
(%o2) --
n
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").
Let \(X_1, X_2, ..., X_n\) be \(n\) independent normally distributed random variables with means \(\mu_k\) and unit variances. Then the random variable
\[\sum_{k=1}^n X_k^2
\]
has a noncentral \(\chi^2\) distribution. The number of degrees of freedom is \(n\), and the noncentrality parameter is defined by
\[\sum_{k=1}^n \mu_k^2
\]
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
\[f(x; n, \lambda) =
{1\over 2}e^{-(x+\lambda)/2} \left(x\over
\lambda\right)^{n/4-1/2}I_{{n\over 2} - 1}\left(\sqrt{n \lambda}\right)
\]
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").
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.
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
\[E[X] = n + \mu \]
where \(\mu\) is the noncentrality parameter ncp.
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
\[V[X] = 2(n+2\mu) \]
where \(\mu\) is the noncentrality parameter ncp.
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
\[D[X] = \sqrt{2(n+2\mu)}
\]
where \(\mu\) is the noncentrality parameter ncp.
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
\[SK[X] = {2^{3/2}(n+3\mu) \over (n+2\mu)^{3/2}}
\]
where \(\mu\) is the noncentrality parameter ncp.
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
\[KU[X] = {12(n+4\mu)\over (2+2\mu)^2}
\]
where \(\mu\) is the noncentrality parameter ncp.
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").
Let \(S_1\) and \(S_2\) be independent random variables with a \(\chi^2\) distribution with degrees of freedom \(n\) and \(m\), respectively. Then
\[F = {S_1/n \over S_2/m}
\] has an \(F\) distribution with \(n\) and \(m\) degrees of
freedom.
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
\[f(x; m, n) =
\cases{
B\left(\displaystyle{m\over 2}, \displaystyle{n\over 2}\right)^{-1}
\left(\displaystyle{m\over n}\right)^{m/ 2}
x^{m/2-1}
\left(1 + \displaystyle{m\over n}x\right)^{-\left(n+m\right)/2} & $x >
0$ \cr
\cr
0 & otherwise
}
\]
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
\[F(x; m, n) =
\cases{
1 - I_z\left(\displaystyle{m\over 2}, {n\over 2}\right) & $x > 0$ \cr
0 & otherwise
}
\]
where
\[z = {n\over mx+n}
\]
and \(I_z(a,b)\) is the beta_incomplete_regularized 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
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
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
\[E[X] = {n\over n-2}
\]
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
\[V[X] = {2n^2(n+m-2) \over m(n-4)(n-2)^2}
\]
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
\[D[X] = {\sqrt{2}\, n \over n-2} \sqrt{n+m-2\over m(n-4)}
\]
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
\[SK[X] = {(n+2m-2)\sqrt{8(n-4)} \over (n-6)\sqrt{m(n+m-2)}}
\]
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
\[KU[X] = 12{m(n+m-2)(5n-22) + (n-4)(n-2)^2 \over m(n-8)(n-6)(n+m-2)}
\]
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
\[F={{n X}\over{m Y}}
\]
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").
The exponential distribution is the probablity distribution of the time between events in a process where the events occur continuously and independently at a constant average rate.
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
\[f(x; m) =
\cases{
me^{-mx} & for $x \ge 0$ \cr
0 & otherwise
}
\]
(%i1) load ("distrib")$
(%i2) pdf_exp(x,m);
- m x
(%o2) m %e unit_step(x)
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
\[F(x; m) =
\cases{
1 - e^{-mx} & $x \ge 0$ \cr
0 & otherwise
}
\]
(%i1) load ("distrib")$
(%i2) cdf_exp(x,m);
- m x
(%o2) (1 - %e ) unit_step(x)
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
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
\[E[X] = {1\over m}
\]
(%i1) load ("distrib")$
(%i2) mean_exp(m);
1
(%o2) -
m
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
\[V[X] = {1\over m^2}
\]
(%i1) load ("distrib")$
(%i2) var_exp(m);
1
(%o2) --
2
m
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
\[D[X] = {1\over m}
\]
(%i1) load ("distrib")$
(%i2) std_exp(m);
1
(%o2) -
m
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
\[SK[X] = 2 \]
(%i1) load ("distrib")$
(%i2) skewness_exp(m); (%o2) 2
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
\[KU[X] = 6 \]
(%i1) load ("distrib")$
(%i2) kurtosis_exp(m); (%o2) 6
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").
The lognormal distribution is distribution for a random variable whose logarithm is normally distributed.
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
\[f(x; m, s) =
\cases{
\displaystyle{1\over x s \sqrt{2\pi}}
\exp\left(-\displaystyle{\left(\log x - m\right)^2\over 2s^2}\right) & for $x \ge
0$ \cr
\cr
0 & for $x < 0$
}
\]
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
\[F(x; m, s) =
\cases{
\displaystyle{1\over 2}\left[1+{\rm erf}\left({\log x - m\over s\sqrt{2}}\right)\right] &
for $x > 0$ \cr
\cr
0 & for $x \le 0$
}
\]
(%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.
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
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
\[E[X] = \exp\left(m+{s^2\over 2}\right)
\]
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
\[V[X] = \left(\exp\left(s^2\right) - 1\right) \exp\left(2m+s^2\right) \]
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
\[D[X] = \sqrt{\left(\exp\left(s^2\right) - 1\right)}
\exp\left(m+{s^2\over 2}\right)
\]
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
\[SK[X] = \left(\exp\left(s^2\right)+2\right)\sqrt{\exp\left(s^2\right)-1}
\]
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
\[KU[X] = \exp\left(4s^2\right)+2\exp\left(3s^2\right)+3\exp\left(2s^2\right)-3 \]
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").
The gamma distribution is a two-parameter family of probability distributions. Maxima uses the parameterization using the shape and scale for the first and second parameters of the distribution.
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
\[f(x; a, b) = {x^{a-1}e^{-x/b}\over b^a \Gamma(a)}
\]
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
\[F(x; a, b) =
\cases{
1-Q(a,{x\over b}) & for $x \ge 0$ \cr
\cr
0 & for $x < 0$
}
\]
where \(Q(a,z)\) is the gamma_incomplete_regularized function.
(%i1) load ("distrib")$
(%i2) cdf_gamma(3,5,21);
1
(%o2) 1 - gamma_incomplete_regularized(5, -)
7
(%i3) float(%); (%o3) 4.402663157376807e-7
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").
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
\[E[X] = ab \]
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
\[V[X] = ab^2 \]
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
\[D[X] = b\sqrt{a}
\]
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
\[SK[X] = {2\over \sqrt{a}}
\]
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
\[KU[X] = {6\over a}
\]
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").
The beta distribution is a family of distributions defined over \([0,1]\) parameterized by two positive shape parameters \(a\), and \(b\).
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
\[f(x; a, b) =
\cases{
\displaystyle{x^{a-1}(1-x)^{b-1} \over B(a,b)} & for $0 \le x \le 1$
\cr
\cr
0 & otherwise
}
\]
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
\[F(x; a, b) =
\cases{
0 & $x < 0$ \cr
I_x(a,b) & $0 \le x \le 1$ \cr
1 & $x > 1$
}
\]
(%i1) load ("distrib")$
(%i2) cdf_beta(1/3,15,2);
11
(%o2) --------
14348907
(%i3) float(%); (%o3) 7.666089131388195e-7
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").
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
\[E[X] = {a\over a+b}
\]
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
\[V[X] = {ab \over (a+b)^2(a+b+1)}
\]
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
\[D[X] = {1\over a+b}\sqrt{ab\over a+b+1}
\]
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
\[SK[X] = {2(b-a)\sqrt{a+b+1} \over (a+b+2)\sqrt{ab}}
\]
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
\[KU[X] = {3(a+b+1)\left(2(a+b)^2+ab(a+b-6)\right) \over
ab(a+b+2)(a+b+3)} - 3
\]
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").
The continuous uniform distribution is constant over the interval \([a,b]\) and is zero elsewhere.
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
\[f(x; a, b) =
\cases{
\displaystyle{1\over b-a} & for $0 \le x \le 1$ \cr
\cr
0 & otherwise
}
\]
and is 0 otherwise.
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
\[F(x; a, b) =
\cases{
0 & for $x < a$ \cr
\cr
\displaystyle{x-a\over b-a} & for $a \le x \le b$ \cr
\cr
1 & for $x > b$
}
\]
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").
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
\[E[X] = {a+b\over 2}
\]
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
\[V[X] = {(b-a)^2\over 12}
\]
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
\[D[X] = {b-a \over 2\sqrt{3}}
\]
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
\[SK[X] = 0 \]
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
\[KU[X] = -{6\over5}
\]
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").
The logistic distribution is a continuous distribution where its cumulative distribution function is the logistic 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
\[f(x; a, b) = {e^{-(x-a)/b} \over b\left(1 + e^{-(x-a)/b}\right)^2}
\]
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
\[F(x; a, b) = {1\over 1+e^{-(x-a)/b}}
\]
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").
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
\[E[X] = a \]
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
\[V[X] = {\pi^2 b^2 \over 3}
\]
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
\[D[X] = {\pi b\over \sqrt{3}}
\]
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
\[SK[X] = 0 \]
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
\[KU[X] = {6\over 5}
\]
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").
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
\[f(x; a, b) =
\cases{
\displaystyle{a b^a \over x^{a+1}} & for $x \ge b$ \cr
\cr
0 & for $x < b$
}
\]
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
\[F(x; a, b) =
\cases{
1-\left(\displaystyle{b\over x}\right)^a & for $x \ge b$\cr
0 & for $x < b$
}
\]
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").
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
\[E[X] = {ab\over a-1}
\]
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
\[V[X] = {ab^2\over (a-2)(a-1)^2}
\]
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
\[D[X] = {b\over a-1} \sqrt{a\over a-2}
\]
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
\[SK[X] = {2(a+1)\over a-3} \sqrt{a-2\over a}
\]
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
\[KU[X] = {6\left(a^3+a^2-6*a-2\right) \over a(a-3)(a-4)} - 3
\]
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").
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
\[f(x; a, b) =
\cases{
\displaystyle{1\over b} \left({x\over b}\right)^{a-1} e^{-(x/b)^a} &
for $x \ge 0$ \cr
\cr
0 & for $x < 0$
}
\]
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
\[F(x; a, b) =
\cases{
1 - e^{-(x/b)^a} & for $x \ge 0$ \cr
0 & for $x < 0$
}
\]
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").
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
\[E[X] = b\Gamma\left(1+{1\over a}\right)
\]
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
\[V[X] = b^2\left[\Gamma\left(1+{2\over a}\right) -
\Gamma\left(1+{1\over a}\right)^2\right]
\]
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
\[D[X] = b\sqrt{\Gamma\left(1+{2\over a}\right) -
\Gamma\left(1+{1\over a}\right)^2}
\]
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
\[SK[X] = {\displaystyle\Gamma\left(1+{3\over a}\right)
-3\Gamma\left(1+{1\over a}\right)\Gamma\left(1+{2\over
a}\right)+2\Gamma\left(1+{1\over a}\right)^3
\over
\displaystyle\left[\Gamma\left(1+{2\over a}\right)-\Gamma\left(1+{1\over
a}\right)^2\right]^{3/2}
}
\]
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
\[KU[X] = {
\Gamma_4
- 4\Gamma_1 \Gamma_3
+ 6\Gamma_1^2 \Gamma_2
- 3 \Gamma_1^4
\over
\left[\Gamma_2 - \Gamma_1^2\right]^2
} - 3
\]
where \(\Gamma_k = \Gamma\left(1+k/a\right).\)
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").
The Rayleigh distribution coincides with the \(\chi^2\) distribution with two degrees of freedom.
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
\[f(x; b) =
\cases{
2b^2 x e^{-b^2 x^2} & for $x \ge 0$ \cr
0 & for $x < 0$
}
\]
(%i1) load ("distrib")$
(%i2) pdf_rayleigh(x,b);
2 2
2 - b x
(%o2) 2 b x %e unit_step(x)
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
\[F(x; b) =
\cases{
1 - e^{-b^2 x^2} & for $x \ge 0$\cr
0 & for $x < 0$
}
\]
(%i1) load ("distrib")$
(%i2) cdf_rayleigh(x,b);
2 2
- b x
(%o2) (1 - %e ) unit_step(x)
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);
2.145966026289347
(%o2) -----------------
b
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
\[E[X] = {\sqrt{\pi}\over 2b}
\]
(%i1) load ("distrib")$
(%i2) mean_rayleigh(b);
sqrt(%pi)
(%o2) ---------
2 b
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
\[V[X] = {1\over b^2}\left(1-{\pi \over 4}\right)
\]
(%i1) load ("distrib")$
(%i2) var_rayleigh(b);
%pi
1 - ---
4
(%o2) -------
2
b
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
\[D[X] = {1\over b}\sqrt{\displaystyle 1 - {\pi\over 4}}
\]
(%i1) load ("distrib")$
(%i2) std_rayleigh(b);
%pi
sqrt(1 - ---)
4
(%o2) -------------
b
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
\[SK[X] = {2\sqrt{\pi}(\pi - 3)\over (4-\pi)^{3/2}}
\]
(%i1) load ("distrib")$
(%i2) skewness_rayleigh(b);
3/2
%pi 3 sqrt(%pi)
------ - -----------
4 4
(%o2) --------------------
%pi 3/2
(1 - ---)
4
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
\[KU[X] = {32-3\pi\over (4-\pi)^2} - 3
\]
(%i1) load ("distrib")$
(%i2) kurtosis_rayleigh(b);
2
3 %pi
2 - ------
16
(%o2) ---------- - 3
%pi 2
(1 - ---)
4
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").
The Laplace distribution is a continuous probability distribution that is sometimes called the double exponential distribution because it can be thought of as two exponential distributions spliced back to back.
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
\[f(x; a, b) = {1\over 2b}\exp\left(-{|x-a|\over b}\right)
\]
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
\[F(x; a, b) =
\cases{
\displaystyle{1\over 2} \exp\left({x-a\over b}\right) & for $x < a$\cr
\cr
1-\displaystyle{1\over 2} \exp\left({x-a\over b}\right) & for $x \ge a$
}
\]
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").
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
\[E[X] = a \]
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
\[V[X] = 2b^2 \]
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
\[D[X] = \sqrt{2} b
\]
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
\[SK[X] = 0 \]
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
\[KU[X] = 3 \]
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").
The Cauchy distribution (also known as the Lorentz distribution) is the distribution of of the ratio of two independent normally distributed random variables with mean zero.
Note that the mean, variance, standard deviation, skewness coefficient, and kurtosis coefficient are all undefined for the Cauchy distribution. The integrals do not converge in this case.
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
\[f(x; a, b) = {b\over \pi\left((x-a)^2+b^2\right)}
\]
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
\[F(x; a, b) = {1\over 2} + {1\over \pi} \tan^{-1} {x-a\over b}
\]
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").
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").
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
\[f(x; a, b) = {1\over b} \exp\left[{a-x\over b} - \exp\left({a-x\over b}\right)\right]
\]
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
\[F(x; a, b) = \exp\left[-\exp\left({a-x\over b}\right)\right]
\]
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").
Returns the mean of a \({\it Gumbel}(a,b)\) random variable, with \(b>0\).
The mean is
\[E[X] = a+b\gamma \]
(%i1) load ("distrib")$
(%i2) mean_gumbel(a,b); (%o2) %gamma b + a
where symbol %gamma stands for the Euler-Mascheroni constant. See also %gamma.
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
\[V[X] = {\pi^2\over 6} b^2
\]
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
\[D[X] = {\pi \over \sqrt{6}} b
\]
Returns the skewness coefficient of a \({\it Gumbel}(a,b)\) random variable, with \(b>0\).
The skewness coefficient is
\[SK[X] = {12\sqrt{6}\over \pi^3} \zeta(3)
\]
(%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.
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
\[KU[X] = {12\over 5}
\]
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").
Maxima knows the following kinds of discrete distributions
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").
(%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
Returns the value at x of the cumulative distribution function of a general finite discrete random variable, with vector probabilities \(v\).
See pdf_general_finite_discrete 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
Returns the q-quantile of a general finite discrete random variable, with vector probabilities \(v\).
See pdf_general_finite_discrete for more details.
Returns the mean of a general finite discrete random variable, with vector probabilities \(v\).
See pdf_general_finite_discrete for more details.
Returns the variance of a general finite discrete random variable, with vector probabilities \(v\).
See pdf_general_finite_discrete for more details.
Returns the standard deviation of a general finite discrete random variable, with vector probabilities \(v\).
See pdf_general_finite_discrete for more details.
Returns the skewness coefficient of a general finite discrete random variable, with vector probabilities \(v\).
See pdf_general_finite_discrete for more details.
Returns the kurtosis coefficient of a general finite discrete random variable, with vector probabilities \(v\).
See pdf_general_finite_discrete for more details.
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.
(%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) [3, 4, 3, 4, 4, 4, 4, 2, 4, 4]
The binomial distribution with parameters \(n\) and \(p\) is a discrete probability distribution. It consists of \(n\) independent experiments where each experiment consists of a Boolean-valued outcome where a success occurs with a probablity \(p\).
For example, a biased coin that comes up heads with probablity \(p\) is tossed \(n\) times. Then the probability of exactly \(k\) heads in \(n\) tosses is given by the binomial distribution.
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
\[f(x; n, p) = {n\choose x} (1-p)^{n-x}p^x
\]
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
\[F(x; n, p) = I_{1-p}(n-\lfloor x \rfloor, \lfloor x \rfloor + 1)
\]
where \(I_z(a,b)\) is the beta_incomplete_regularized function.
(%i1) load ("distrib")$
(%i2) cdf_binomial(5,7,1/6);
7775
(%o2) ----
7776
(%i3) float(%); (%o3) 0.9998713991769548
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").
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
\[E[X] = np \]
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
\[V[X] = np(1-p) \]
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
\[D[X] = \sqrt{np(1-p)}
\]
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
\[SK[X] = {1-2p\over \sqrt{np(1-p)}}
\]
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
\[KU[X] = {1-6p(1-p)\over np(1-p)}
\]
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").
The Poisson distribution is a discrete probability distribution. It is the probability that a given number of events occur in a fixed interval when the events occur independently of the time of the last event, and the events occur with a known constant rate.
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
\[f(x; m) = {m^x e^{-m}\over x!}
\]
Returns the value at x of the cumulative distribution function of a \({\it Poisson}(m)\) random variable, with \(m>0\).
The cdf is
\[F(x; m) = Q(\lfloor x \rfloor + 1, m) \]
where \(Q(x,m)\) is the gamma_incomplete_regularized function.
(%i1) load ("distrib")$
(%i2) cdf_poisson(3,5); (%o2) gamma_incomplete_regularized(4, 5)
(%i3) float(%); (%o3) 0.2650259152973619
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").
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
\[E[X] = m \]
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
\[V[X] = m \]
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
\[V[X] = \sqrt{m}
\]
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
\[SK[X] = {1\over \sqrt{m}}
\]
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
\[KU[X] = {1\over m}
\]
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").
The Bernoulli distribution is a discrete probability distribution which takes on two values, 0 and 1. The value 1 occurs with probability \(p\), and 0 occurs with probabilty \(1-p\).
It is equivalent to the \({\it Binomial}(1,p)\) distribution (see Binomial Random Variable)
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
\[f(x; p) = p^x (1-p)^{1-x}
\]
(%i1) load ("distrib")$
(%i2) pdf_bernoulli(1,p); (%o2) p
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
\[F(x; p) = I_{1-p}(1-\lfloor x \rfloor, \lfloor x \rfloor + 1)
\]
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").
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
\[E[X] = p \]
(%i1) load ("distrib")$
(%i2) mean_bernoulli(p); (%o2) p
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
\[V[X] = p(1-p) \]
(%i1) load ("distrib")$
(%i2) var_bernoulli(p); (%o2) (1 - p) p
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
\[D[X] = \sqrt{p(1-p)}
\]
(%i1) load ("distrib")$
(%i2) std_bernoulli(p); (%o2) sqrt((1 - p) p)
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
\[SK[X] = {1-2p \over \sqrt{p(1-p)}}
\]
(%i1) load ("distrib")$
(%i2) skewness_bernoulli(p);
1 - 2 p
(%o2) ---------------
sqrt((1 - p) p)
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
\[KU[X] = {1-6p(1-p) \over p(1-p)}
\]
(%i1) load ("distrib")$
(%i2) kurtosis_bernoulli(p);
1 - 6 (1 - p) p
(%o2) ---------------
(1 - p) p
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").
The Geometric distibution is a discrete probability distribution. It is the distribution of the number Bernoulli trials that fail before the first success.
Consider flipping a biased coin where heads occurs with probablity \(p\). Then the probability of \(k-1\) tails in a row followed by heads is given by the \({\it Geometric}(p)\) distribution.
Returns the value at x of the probability function of a \({\it Geometric}(p)\) random variable, with \(0 < p \leq 1\)
The pdf is
\[f(x; p) = p(1-p)^x \]
This is interpreted as the probability of \(x\) failures before the first success.
load("distrib") loads this 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
\[1-(1-p)^{1 + \lfloor x \rfloor}
\]
load("distrib") loads this 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.
Returns the mean of a \({\it Geometric}(p)\) random variable, with \(0 < p \leq 1\).
The mean is
\[E[X] = {1\over p} - 1
\]
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.
Returns the variance of a \({\it Geometric}(p)\) random variable, with \(0 < p \leq 1\).
The variance is
\[V[X] = {1-p\over p^2}
\]
load("distrib") loads this function.
Returns the standard deviation of a \({\it Geometric}(p)\) random variable, with \(0 < p \leq 1\).
\[D[X] = {\sqrt{1-p} \over p}
\]
load("distrib") loads this function.
Returns the skewness coefficient of a \({\it Geometric}(p)\) random variable, with \(0 < p \leq 1\).
The skewness coefficient is
\[SK[X] = {2-p \over \sqrt{1-p}}
\]
load("distrib") loads this function.
Returns the kurtosis coefficient of a geometric random variable \({\it Geometric}(p)\) , with \(0 < p \leq 1\).
The kurtosis coefficient is
\[KU[X] = {p^2-6p+6 \over 1-p}
\]
load("distrib") loads this 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.
The Discrete uniform distribution is a discrete probablity distribution where a finite number of values are equally likely to occur. The values are \(1,2,3,...,n\).
For example throwing a fair die of 6 sides numbered 1 through 6 follows a \({\it DiscreteUniform}(1/6)\) distribution.
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
\[f(x,n) = {1\over n}
\]
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
\[F(x; n) = {\lfloor x \rfloor \over n}
\]
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").
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
\[E[X] = {n+1\over 2}
\]
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
\[V[X] = {n^2-1 \over 12}
\]
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
\[D[X] = {\sqrt{n^2-1} \over 2\sqrt{3}}
\]
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
\[SK[X] = 0 \]
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
\[KU[X] = - {6(n^2+1)\over 5 (n^2-1)}
\]
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").
The hypergeometric distribution is a discrete probability distribution.
Let \(n_1\) be the number of objects of a class \(A\) and \(n_2\) be the number of objects of class \(B\). We take out \(n\) objects, without replacment. Then the hypergeometric distribution is the probability that exactly \(k\) objects are from class \(A\). Of course \(n \leq n_1 + n_2\).
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
\[f(x; n_1, n_2, n) = {\displaystyle{n_1\choose x} {n_2 \choose n-x}
\over \displaystyle{n_2+n_1 \choose n}}
\]
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
\[F(x; n_1, n_2, n) = {n_2+n_1\choose n}^{-1}
\sum_{k=0}^{\lfloor x \rfloor} {n_1 \choose k} {n_2 \choose n - k}
\]
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").
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
\[E[X] = {n n_1\over n_2+n_1}
\]
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
\[V[X] = {n n_1 n_2 (n_1 + n_2 - n)
\over
(n_1 + n_2 - 1) (n_1 + n_2)^2}
\]
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
\[D[X] = {1\over n_1+n_2}\sqrt{n n_1 n_2 (n_1 + n_2 - n) \over n_1+n_2-1}
\]
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
\[SK[X] = {(n_2-n_2)(n_1+n_2-2n)\over n_1+n_2-2}
\sqrt{n_1+n_2-1 \over n n_1 n_2 (n_1+n_2-n)}
\]
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
\[
\eqalign{
KU[X] = &
\left[{C(1)C(0)^2
\over
n n_1 n_2 C(3)C(2)C(n)}\right. \cr
& \times
\left.\left(
{3n_1n_2\left((n-2)C(0)^2+6nC(n)-n^2C(0)\right)
\over
C(0)^2
}
-6nC(n) + C(0)C(-1)
\right)\right] \cr
&-3
}
\]
where \(C(k) = n_1+n_2-k.\)
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").
The negative binomial distribution is a discrete probability distribution. Suppose we have a sequence of Bernoulli trials where each trial has two outcomes called “success” and “failure” where “success” occurs with probablity \(p\) and “failure” with probability \(1-p\). We observe the sequence until a predefined number \(r\) of sucesses have occurred. Then the number of failures seen will have a \({\it NegativeBinomial}(r,p)\) distribution.
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
\[f(x; n, p) = {x+n-1 \choose n-1} (1-p)^xp^n
\]
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
\[F(x; n, p) = I_p(n,\lfloor x \rfloor + 1) \]
where \(I_p(a,b)\) is the beta_incomplete_regularized function.
(%i1) load ("distrib")$
(%i2) cdf_negative_binomial(3,4,1/8);
3271
(%o2) ------
524288
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").
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
\[E[X] = {n(1-p)\over p}
\]
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
\[V[X] = {n(1-p)\over p^2}
\]
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
\[D[X] = {\sqrt{n(1-p)}\over p}
\]
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
\[SK[X] = {2-p \over \sqrt{n(1-p)}}
\]
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
\[KU[X] = {p^2-6p+6 \over n(1-p)}
\]
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").