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:withStatistics:
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:=RandomVariableChiSquare5
C:=_R
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.
PDFC,x
{0x<0162x3/2ⅇ−12xπotherwise
PDFChiSquare5,x
PDFC,x,mainbranch
162x3/2ⅇ−12xπ
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:=RandomVariableUniform0,1RandomVariableNormal0,1
R:=_R1_R2
PDFR,x
14Ei1,12x22π
CDFR,x
142π+2xEi1,12x2+2erf12x2ππ
VarianceR
13
ExpectedValuecosR
122πerf122
2 Discrete Probability Distributions
The Statistics package includes 9 discrete probability distributions and commands for manipulating and creating discrete random variables.
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:=RandomVariableBinomial4,0.5
R:=_R
ProbabilityFunctionR,x
{0x<0binomial4,x0.5x0.54−xotherwise
ProbabilityFunctionBinomial4,0.5,x
1.00
MomentR,2
5.0000
MomentGeneratingFunctionR,x
0.5ⅇx+0.54
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:=RandomVariableNegativeBinomial2,34
T:=_R1
CDFT,x assuming x ∷ posint
−7164−x−3164−xx+1
CDFT,4
40774096
MeanT
23
CentralMomentT,3
4027
CumulantGeneratingFunctionT,x
2ln34−2ln1−14ⅇIx
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:=12,14,18,116,116:P≔RandomVariableProbabilityTablePTable:
ProbabilityFunctionP,2
14
CDFP,2
34
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:=Array1,1,1,2,4,4,5.5:X≔RandomVariableEmpiricalDistributionEmpiricalData:
ProbabilityFunctionX,1
37
CDFX,5
67
MeanX
2.64285714285714279
CentralMomentX,3
2.10787172011661861
3 Random Sample Generation
All probability distributions provide optimized hardware-level random number generators capable of generating very large pseudo-random samples quickly.
Generate a sample from a Binomial distribution.
R:=RandomVariableBinomial10,0.5:SampleR,10
Generate a sample from a probability table distribution.
PTable:=12,14,18,116,116:P:=RandomVariableProbabilityTablePTable:SampleP,10
Sample a non-central chi square distribution and plot the histogram of the output against the probability density function.
S:=SampleNonCentralChiSquare5,5,100000:P1≔HistogramS,range=0..30,maxbins=100:P2:=DensityPlotNonCentralChiSquare5,5,range=0..30,thickness=3,color=red:plotsdisplayP1,P2
4 Custom Random Variables
The Statistics package includes the Distribution constructor, which can be used to create custom random variables.
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:=PDFNormal0,1,x
NormalPDF:=122ⅇ−12x2π
HalfNormalPDF:=piecewisex<0,0,2⋅NormalPDF
HalfNormalPDF:={0x<02ⅇ−12x2πotherwise
Create a distribution module using the half normal PDF.
HalfNormalDist:=DistributionPDF=unapplyHalfNormalPDF,x
HalfNormalDist:=moduleoptionDistribution,Continuous;exportPDF,Type;end module
R:=RandomVariableHalfNormalDist
R:=_R0
DensityPlotR,thickness=3
Compute the characteristics of this distribution.
MeanR
2π
−2+ππ
erf12x2ⅇ12x2+ⅇ12x2
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