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Probability Distributions

The Statistics package contains 37 probability distributions as well as providing functionality for creating new distributions and manipulating random variables.

1 Continuous Probability Distributions

The Statistics package includes 28 continuous probability distributions along with commands for manipulating and creating continuous random variables. Continuous probability distributions are defined by a continuous probability density function along a section of the real line.

$restart &colon;$ $with (Statistics) &colon;$

Consider a chi square random variable. The chi square random variable takes a single parameter which represents the number of degrees of freedom. When the random variable is created using the RandomVariable constructor, it generates a new name for the random variable data structure and returns it.

$C := RandomVariable (ChiSquare (5))$

$C :=_R$

(1.1)

The probability density function, as well as all other distribution commands, accepts either a random variable or probability distribution as its first parameter. The 'mainbranch' option can be used to return only the main branch of the distribution.

$PDF (C, x)$

${\begin{matrix} 0 & x < 0 \\ \frac{1}{6} \frac{\sqrt{2} x^{3 / 2} {&ExponentialE;}^{- \frac{1}{2} x}}{\sqrt{π}} & otherwise \end{matrix}$

(1.2)

$PDF (ChiSquare (5), x)$

${\begin{matrix} 0 & x < 0 \\ \frac{1}{6} \frac{\sqrt{2} x^{3 / 2} {&ExponentialE;}^{- \frac{1}{2} x}}{\sqrt{π}} & otherwise \end{matrix}$

(1.3)

$PDF (C, x, mainbranch)$

$\frac{1}{6} \frac{\sqrt{2} x^{3 / 2} {&ExponentialE;}^{- \frac{1}{2} x}}{\sqrt{π}}$

(1.4)

Combinations of probability distributions can be generated by performing operations on a set of random variables. For example, consider the product of a uniform random variable and a normal (gaussian) random variable.

$R := RandomVariable (Uniform (0, 1)) RandomVariable (Normal (0, 1))$

$R :=_R1_R2$

(1.5)

$PDF (R, x)$

$\frac{1}{4} \frac{Ei (1, \frac{1}{2} x^{2}) \sqrt{2}}{\sqrt{π}}$

(1.6)

$CDF (R, x)$

$\frac{1}{4} \frac{2 \sqrt{π} + \sqrt{2} x Ei (1, \frac{1}{2} x^{2}) + 2 \erf (\frac{1}{2} x \sqrt{2}) \sqrt{π}}{\sqrt{π}}$

(1.7)

$Variance (R)$

$\frac{1}{3}$

(1.8)

$ExpectedValue (\cos (R))$

$\frac{1}{2} \sqrt{2} \sqrt{π} \erf (\frac{1}{2} \sqrt{2})$

(1.9)

2 Discrete Probability Distributions

The Statistics package includes 9 discrete probability distributions and commands for manipulating and creating discrete random variables.

$restart &colon;$ $with (Statistics) &colon;$

Consider a binomial random variable. Unlike continuous random variables, discrete random variables are defined by their probability function rather than their probability density function.

$R := RandomVariable (Binomial (4, 0.5))$

$R :=_R$

(2.1)

$ProbabilityFunction (R, x)$

${\begin{matrix} 0 & x < 0 \\ binomial (4, x) {0.5}^{x} {0.5}^{4 - x} & otherwise \end{matrix}$

(2.2)

$ProbabilityFunction (Binomial (4, 0.5), x)$

${\begin{matrix} 0 & x < 0 \\ binomial (4, x) {0.5}^{x} {0.5}^{4 - x} & otherwise \end{matrix}$

(2.3)

$Variance (R)$

$1.00$

(2.4)

$Moment (R, 2)$

$5.0000$

(2.5)

$MomentGeneratingFunction (R, x)$

${(0.5 {&ExponentialE;}^{x} + 0.5)}^{4}$

(2.6)

The Statistics package also allows for both numeric and symbolic manipulation of random variables and distributions. Consider the negative binomial distribution with symbolic parameters.

$T := RandomVariable (NegativeBinomial (2, \frac{3}{4}))$

$T :=_R1$

(2.7)

$CDF (T, x) assuming x ∷ posint$

$- \frac{7}{16} 4^{- x} - \frac{3}{16} 4^{- x} x + 1$

(2.8)

$CDF (T, 4)$

$\frac{4077}{4096}$

(2.9)

$Mean (T)$

$\frac{2}{3}$

(2.10)

$CentralMoment (T, 3)$

$\frac{40}{27}$

(2.11)

$CumulantGeneratingFunction (T, x)$

$2 \ln (\frac{3}{4}) - 2 \ln (1 - \frac{1}{4} {&ExponentialE;}^{I x})$

(2.12)

Further, the Statistics package supports the probability table. This distribution is used to associate probabilities with the integers 1..n, for any n. Consider a case of n = 5.

$PTable := [\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \frac{1}{16}] &colon;$ $P ≔ RandomVariable (ProbabilityTable (PTable)) &colon;$

$ProbabilityFunction (P, 2)$

$\frac{1}{4}$

(2.13)

$CDF (P, 2)$

$\frac{3}{4}$

(2.14)

The Statistics package also supports the empirical distribution, which is effectively a probability distribution built around a data sample. The probability of each element is equal to its frequency in the data sample.

$EmpiricalData := Array ([1, 1, 1, 2, 4, 4, 5.5]) &colon;$ $X ≔ RandomVariable (EmpiricalDistribution (EmpiricalData)) &colon;$

$ProbabilityFunction (X, 1)$

$\frac{3}{7}$

(2.15)

$CDF (X, 5)$

$\frac{6}{7}$

(2.16)

$Mean (X)$

$2.64285714285714279$

(2.17)

$CentralMoment (X, 3)$

$2.10787172011661861$

(2.18)

3 Random Sample Generation

All probability distributions provide optimized hardware-level random number generators capable of generating very large pseudo-random samples quickly.

$restart &colon;$ $with (Statistics) &colon;$

Generate a sample from a Binomial distribution.

$R := RandomVariable (Binomial (10, 0.5)) &colon;$ $Sample (R, 10)$

Generate a sample from a probability table distribution.

$PTable := [\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \frac{1}{16}] &colon;$ $P := RandomVariable (ProbabilityTable (PTable)) &colon;$ $Sample (P, 10)$

Sample a non-central chi square distribution and plot the histogram of the output against the probability density function.

$S := Sample (NonCentralChiSquare (5, 5), 100000) &colon;$ $P1 ≔ Histogram (S, range = 0 .. 30, maxbins = 100) &colon;$ $P2 := DensityPlot (NonCentralChiSquare (5, 5), range = 0 .. 30, thickness = 3, color = red) &colon;$ $plots [display] (P1, P2)$

4 Custom Random Variables

The Statistics package includes the Distribution constructor, which can be used to create custom random variables.

$restart &colon;$ $with (Statistics) &colon;$

A distribution that is occasionally used in statistics is the half-normal distribution, named so because it is a normal distribution that has been cropped at all negative values.

$NormalPDF := PDF (Normal (0, 1), x)$

$NormalPDF := \frac{1}{2} \frac{\sqrt{2} {&ExponentialE;}^{- \frac{1}{2} x^{2}}}{\sqrt{π}}$

(4.1)

$HalfNormalPDF := piecewise (x < 0, 0, 2 \cdot NormalPDF)$

$HalfNormalPDF := {\begin{matrix} 0 & x < 0 \\ \frac{\sqrt{2} {&ExponentialE;}^{- \frac{1}{2} x^{2}}}{\sqrt{π}} & otherwise \end{matrix}$

(4.2)

Create a distribution module using the half normal PDF.

$HalfNormalDist := Distribution (PDF = unapply (HalfNormalPDF, x))$

$HalfNormalDist := module () option Distribution, Continuous &semi; export PDF, Type &semi; end module$

(4.3)

$R := RandomVariable (HalfNormalDist)$

$R :=_R0$

(4.4)

$DensityPlot (R, thickness = 3)$

Compute the characteristics of this distribution.

$Mean (R)$

$\frac{\sqrt{2}}{\sqrt{π}}$

(4.5)

$Variance (R)$

$\frac{- 2 + π}{π}$

(4.6)

$MomentGeneratingFunction (R, x)$

$\erf (\frac{1}{2} x \sqrt{2}) {&ExponentialE;}^{\frac{1}{2} x^{2}} + {&ExponentialE;}^{\frac{1}{2} x^{2}}$

(4.7)

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