The Distribution command creates a new distribution with the specified distribution, functions, or quantities.

The first parameter can be one of the supported distributions or a distribution function or quantity.

A distribution can be defined by the following functions and quantities. The other functions and quantities are calculated if possible.  If not specified otherwise, the default value for all of these functions and quantities is FAIL, indicating that this quantity needs to be computed from others.

CDF - operator

Operator mapping a value $t$ to the CDF of this distribution at $t$.

CDFNumeric - appliable

Procedure for efficiently computing a floating-point approximation to the CDF.

CentralMoment - operator

Operator mapping a positive integer $n$ to the $n$th CentralMoment of this distribution.

CFNumeric - appliable

Procedure for efficiently computing a floating-point approximation to the CharacteristicFunction.

CGF - operator

Operator mapping a value $t$ to the CGF of this distribution at $t$.

CGFNumeric - appliable

Procedure for efficiently computing a floating-point approximation to the CGF.

CharacteristicFunction - operator

Operator mapping a value $t$ to the CharacteristicFunction of this distribution at $t$.

Conditions - list(boolean)

List of conditions on parameters occurring in the other defining equations for this distribution. The conditions need to be satisfied for the distribution to be valid. If the conditions are at any point known not to be satisfied, an error is raised; otherwise, the conditions are implicitly assumed to be satisfied. For this quantity, the default value is the empty list, []; you cannot specify FAIL.

If infolevel for Statistics is set to at least $2$, the implicit assumptions are printed when they are made.

Cumulant - operator

Operator mapping a positive integer $n$ to the $n$th Cumulant of this distribution.

DiscreteValueMap - operator

See the DiscreteValueMap help page.

HazardRate - operator

Operator mapping a value $t$ to the HazardRate of this distribution at $t$.

HodgesLehmann - algebraic

Expression giving the Hodges-Lehmann estimator of this distribution.

InverseSurvivalFunction - operator

Operator mapping a value $t$ to the InverseSurvivalFunction of this distribution at $t$.

ISFNumeric - appliable

Procedure for efficiently computing a floating-point approximation to the InverseSurvivalFunction.

Kurtosis - algebraic

Expression giving the Kurtosis of this distribution.

Mean - algebraic

Expression giving the Mean of this distribution.

Median - algebraic

Expression giving the Median of this distribution.

MGF - operator

Operator mapping a value $t$ to the MGF of this distribution at $t$.

MGFNumeric - appliable

Procedure for efficiently computing a floating-point approximation to the MGF.

MillsRatio - operator

Operator mapping a value $t$ to the MillsRatio of this distribution at $t$.

Mode - algebraic

Expression giving the Mode of this distribution.

Moment - operator

Operator mapping a positive integer $n$ to the $n$th Moment of this distribution.

PDF - operator

Operator mapping a value $t$ to the PDF of this distribution at $t$.

PDFNumeric - appliable

Procedure for efficiently computing a floating-point approximation to the PDF.

PFNumeric - appliable

Procedure for efficiently computing a floating-point approximation to the ProbabilityFunction.

ProbabilityFunction - operator

Operator mapping a value $t$ to the ProbabilityFunction of this distribution at $t$.

Quantile - operator

Operator mapping a value $t$ between 0 and 1 to the $t$th Quantile of this distribution.

QuantileNumeric - appliable

Procedure for efficiently computing a floating-point approximation to the Quantile.

RandomSample - appliable

Procedure for efficiently computing a random sample of this distribution. The procedure should accept all of the calling sequences that Sample accepts:

When called with a non-negative integer argument $n$, the procedure should return the sample as a Vector of length $n$ having datatype = float.

When called with an integer range or a list of integer ranges, the procedure should return the sample as an Array with those dimensions having datatype = float.

When called with any rtable having datatype = float, the procedure should return the sample in that rtable.

RousseeuwCrouxQn - algebraic

Expression giving Rousseeuw and Croux' Qn of this distribution.

RousseeuwCrouxSn - algebraic

Expression giving Rousseeuw and Croux' Sn of this distribution.

Skewness - algebraic

Expression giving the Skewness of this distribution.

Specialize - appliable

Procedure returning the distribution that Specialize of this distribution will return. The procedure should take a list of equations as its single argument, and return a new distribution data structure; the most convenient way to produce one is to call Distribution itself.

StandardDeviation - algebraic

Expression giving the StandardDeviation of this distribution.

StandardizedMoment - operator

Operator mapping a positive integer $n$ to the $n$th StandardizedMoment of this distribution.

SFNumeric - appliable

Procedure for efficiently computing the SurvivalFunction of this distribution.

Support - algebraic .. algebraic

Range containing the Support of this distribution. If a DiscreteValueMap is specified, then this should be the support in the source domain. See the DiscreteValueMap help page for more details. If the Support is not given explicitly, an attempt is made to infer the Support from the PDF or the ProbabilityFunction, falling back to the default value of $-\mathrm{\infty }..\mathrm{\infty }$ if this attempt fails.

SurvivalFunction - operator

Operator mapping a value $t$ to the SurvivalFunction of this distribution at $t$.

Type - equal to either discrete or continuous

Indicates whether the distribution is discrete (meaning it can only assume finitely or countably many values, each with positive probability) or continuous (meaning the probability for it to assume any particular value is $0$). The default value for this property is continuous; it cannot be FAIL. For discrete distributions that can assume non-integer values, you will need to specify a DiscreteValueMap. See the DiscreteValueMap help page for more details.

Variance - algebraic

Expression giving the Variance of this distribution.

$\mathrm{with}\left(\mathrm{Statistics}\right)\:$

$G≔\mathrm{Distribution}\left(\mathrm{GammaDistribution}\left(a\,b\right)\right)$

$G≔\mathbf{module}\left(\right)\mathbf{option}\mathrm{Distribution}\,\mathrm{Continuous}\;\mathbf{export}\mathrm{Conditions}\,\mathrm{Dimensions}\,\mathrm{ParentName}\,\mathrm{Parameters}\,\mathrm{CharacteristicFunction}\,\mathrm{CGF}\,\mathrm{Mean}\,\mathrm{Mode}\,\mathrm{MGF}\,\mathrm{PDF}\,\mathrm{Support}\,\mathrm{Variance}\,\mathrm{CDFNumeric}\,\mathrm{QuantileNumeric}\,\mathrm{RandomSample}\,\mathrm{RandomSampleSetup}\,\mathrm{RandomVariate}\,\mathrm{MaximumLikelihoodEstimate}\;\mathbf{end\; module}$

$Y≔\mathrm{RandomVariable}\left(G\right)$

$Y≔\mathrm{\_R}$

$\mathrm{PDF}\left(Y\,c\right)$

$\left\{\begin{array}{cc}0& c<0\\ \frac{{\left(\frac{c}{a}\right)}^{b-1}{\&ExponentialE;}^{-\frac{c}{a}}}{a\mathrm{\Gamma }\left(b\right)}& \mathrm{otherwise}\end{array}\right.$

$U≔\mathrm{Distribution}\left(\mathrm{`=`}\left(\mathrm{PDF}\&comma;t\mapsto f\left(t\right)\right)\&comma;\mathrm{`=`}\left(\mathrm{CDF}\&comma;t\mapsto F\left(t\right)\right)\&comma;\mathrm{Mean}=3\right)$

$U≔\mathbf{module}\left(\right)\mathbf{option}\mathrm{Distribution}\&comma;\mathrm{Continuous}\&semi;\mathbf{export}\mathrm{CDF}\&comma;\mathrm{Conditions}\&comma;\mathrm{PDF}\&comma;\mathrm{Mean}\&semi;\mathbf{end\; module}$

$Z≔\mathrm{RandomVariable}\left(U\right)$

$Z≔\mathrm{\_R0}$

$\mathrm{PDF}\left(Z\&comma;t\right)$

$f\left(t\right)$

$\mathrm{CDF}\left(Z\&comma;t\right)$

$F\left(t\right)$

$\mathrm{MGF}\left(Z\&comma;t\right)$

${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\&ExponentialE;}^{\mathrm{\_t}t}f\left(\mathrm{\_t}\right)\&DifferentialD;\mathrm{\_t}$

$\mathrm{Mean}\left(Z\right)$

$3$

$\mathrm{Moment}\left(Z\&comma;1\right)$

$3$

$\mathrm{Moment}\left(Z\&comma;2\right)$

${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{\_t0}}^{2}f\left(\mathrm{\_t0}\right)\&DifferentialD;\mathrm{\_t0}$

$\mathrm{myPDF}≔\mathrm{unapply}\left(\mathrm{PDF}\left(\mathrm{BetaDistribution}\left(1\&comma;3\right)\&comma;t\right)\&comma;t\right)$

$\mathrm{myPDF}≔t\mapsto \left\{\begin{array}{cc}0& t<0\\ 3\cdot {\left(1-t\right)}^2& t<1\\ 0& \mathrm{otherwise}\end{array}\right.$

myCDFNumeric := proc(t)
 return evalf(piecewise(t<0, 0, t<1, 3*t*hypergeom([-2, 1], [2], t), 1));
end proc;
trace(myCDFNumeric):

$\mathrm{myCDFNumeric}≔\mathbf{proc}\left(t\right)\mathbf{return}\mathrm{evalf}\left(\mathrm{piecewise}\left(t<0\&comma;0\&comma;t<1\&comma;3\&ast;t\&ast;\mathrm{hypergeom}\left(\left[\&minus;2\&comma;1\right]\&comma;\left[2\right]\&comma;t\right)\&comma;1\right)\right)\mathbf{end\; proc}$

$\mathrm{dist}≔\mathrm{Distribution}\left(\mathrm{PDF}=\mathrm{myPDF}\&comma;\mathrm{CDFNumeric}=\mathrm{myCDFNumeric}\right)$

$\mathrm{dist}≔\mathbf{module}\left(\right)\mathbf{option}\mathrm{Distribution}\&comma;\mathrm{Continuous}\&semi;\mathbf{export}\mathrm{Conditions}\&comma;\mathrm{CDFNumeric}\&comma;\mathrm{PDF}\&semi;\mathbf{end\; module}$

$X≔\mathrm{RandomVariable}\left(\mathrm{dist}\right)$

$X≔\mathrm{\_R2}$

$\mathrm{CDF}\left(X\&comma;\frac{1}{2}\right)$

$\frac{7}{8}$

$\mathrm{CDF}\left(X\&comma;0.5\right)$

{--> enter q, args = .5
<-- exit q (now in GetValue) = .8749999998}

$0.8749999998$

$\mathrm{CDF}\left(X\&comma;\frac{1}{2}\&comma;'\mathrm{numeric}'\right)$

{--> enter q, args = 1/2
<-- exit q (now in GetValue) = .8750000000}

$0.8750000000$

$\mathrm{Median}\left(X\right)$

$-\frac{2^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}{2}+1$

$\mathrm{Median}\left(X\&comma;'\mathrm{numeric}'\right)$

{--> enter q, args = 0.
<-- exit q (now in extproc) = 0.}
{--> enter q, args = .25
<-- exit q (now in extproc) = .578124999999975}
{--> enter q, args = .25
<-- exit q (now in extproc) = .578124999999975}
{--> enter q, args = 0.
<-- exit q (now in extproc) = 0.}
{--> enter q, args = .25
<-- exit q (now in extproc) = .578124999999975}
{--> enter q, args = .216216216216226
<-- exit q (now in extproc) = .518508281839222}
{--> enter q, args = .206102270110636
<-- exit q (now in extproc) = .499627215365731}
{--> enter q, args = .206301958179384
<-- exit q (now in extproc) = .500004694760124}
{--> enter q, args = .206299474633007
<-- exit q (now in extproc) = .50000000116623}
{--> enter q, args = .206299474015915
<-- exit q (now in extproc) = .500000000000049}
{--> enter q, args = .206299474015889
<-- exit q (now in extproc) = .499999999999986}

$0.206299474015889$

$\mathrm{parameterizedPDF}≔\mathrm{unapply}\left(\mathrm{piecewise}\left(t<0\&comma;0\&comma;t<\mathrm{root}\left(\frac{9}{4a}\&comma;3\right)\&comma;\mathrm{sqrt}\left(at\right)\&comma;0\right)\&comma;t\right)$

$\mathrm{parameterizedPDF}≔t\mapsto \left\{\begin{array}{cc}0& t<0\\ \sqrt{a\cdot t}& t<\frac{{18}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\cdot {\left(\frac{1}{a}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}{2}\\ 0& \mathrm{otherwise}\end{array}\right.$

$\mathrm{parameterized}≔\mathrm{Distribution}\left(\mathrm{PDF}=\mathrm{parameterizedPDF}\&comma;\mathrm{Conditions}=\left[0<a\right]\right)$

$\mathrm{parameterized}≔\mathbf{module}\left(\right)\mathbf{option}\mathrm{Distribution}\&comma;\mathrm{Continuous}\&semi;\mathbf{export}\mathrm{Conditions}\&comma;\mathrm{PDF}\&semi;\mathbf{end\; module}$

$\mathrm{infolevel}\left[\mathrm{Statistics}\right]≔2$

${\mathrm{infolevel}}_{\mathrm{Statistics}}≔2$

$\mathrm{PDF}\left(\mathrm{parameterized}\&comma;t\right)$

Statistics:-PDF -- using the following implicit assumptions: {0 < a}

$\left\{\begin{array}{cc}0& t<0\\ \sqrt{at}& t<\frac{{18}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}{\left(\frac{1}{a}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}{2}\\ 0& \mathrm{otherwise}\end{array}\right.$

$\mathrm{kurtosis}≔\mathrm{Kurtosis}\left(\mathrm{parameterized}\right)$

Statistics:-Kurtosis -- using the following implicit assumptions: {0 < a}

$\mathrm{kurtosis}≔\frac{14\left(122872^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}{3}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}-11823{18}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\right)}{11{\left(632^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}{3}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}-25{18}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\right)}^2}$

$\mathrm{simplify}\left(\mathrm{kurtosis}\right)$

$\frac{203}{99}$

$\mathrm{assume}\left(0<m\&comma;0<k\&comma;0<T\right)$

$\mathrm{\sigma }≔\mathrm{sqrt}\left(\frac{kT}{m}\right)$

$\mathrm{\sigma }≔\sqrt{\frac{\mathrm{k~}\mathrm{T~}}{\mathrm{m~}}}$

$f≔\mathrm{simplify}\left(\mathrm{piecewise}\left(x<0\&comma;0\&comma;\frac{\mathrm{sqrt}\left(\frac{2}{\mathrm{\pi }}\right)}{{\mathrm{\sigma }}^3}{x}^2\exp \left(-\frac{x^2}{2{\mathrm{\sigma }}^2}\right)\right)\right)$

$f≔\left\{\begin{array}{cc}0& x<0\\ \frac{\sqrt{2}{\mathrm{m~}}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{x}^2{\&ExponentialE;}^{-\frac{x^2\mathrm{m~}}{2\mathrm{k~}\mathrm{T~}}}}{\sqrt{\mathrm{\pi }}{\mathrm{k~}}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\mathrm{T~}}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}& 0\le x\end{array}\right.$

$\mathrm{MD}≔\mathrm{Distribution}\left(\mathrm{PDF}=\mathrm{unapply}\left(f\&comma;x\right)\right)$

$\mathrm{MD}≔\mathbf{module}\left(\right)\mathbf{option}\mathrm{Distribution}\&comma;\mathrm{Continuous}\&semi;\mathbf{export}\mathrm{Conditions}\&comma;\mathrm{PDF}\&semi;\mathbf{end\; module}$

$X≔\mathrm{RandomVariable}\left(\mathrm{MD}\right)$

$X≔\mathrm{\_R5}$

$\mathrm{Mean}\left(X\right)$

$\frac{2\sqrt{\mathrm{T~}}\sqrt{\mathrm{k~}}\sqrt{2}}{\sqrt{\mathrm{m~}}\sqrt{\mathrm{\pi }}}$

$\mathrm{Mean}\left(\mathrm{Maxwell}\left(\mathrm{\sigma }\right)\right)$

$\frac{2\sqrt{2}\sqrt{\frac{\mathrm{k~}\mathrm{T~}}{\mathrm{m~}}}}{\sqrt{\mathrm{\pi }}}$

$\mathrm{simplify}\left(\mathrm{ExpectedValue}\left(\frac{1}{2}m{X}^2\right)\right)$

$\frac{3\mathrm{k~}\mathrm{T~}}{2}$

$g≔\mathrm{eval}\left(f\&comma;\left[T=298.15\&comma;k=\mathrm{evalf}\left(\mathrm{ScientificConstants}:-\mathrm{Constant}\left('k'\right)\right)\&comma;m=4.\times {10}^{\mathrm{-27}}\right]\right)$

$g≔\left\{\begin{array}{cc}0& x<0\\ 5.404283672\times {10}^{\mathrm{-10}}\sqrt{2}{x}^2{\&ExponentialE;}^{-4.858610154\times {10}^{\mathrm{-7}}{x}^2}& 0\le x\end{array}\right.$

$\mathrm{\alpha }≔\mathrm{eval}\left(\mathrm{\sigma }\&comma;\left[T=298.15\&comma;k=\mathrm{evalf}\left(\mathrm{ScientificConstants}:-\mathrm{Constant}\left('k'\right)\right)\&comma;m=4.\times {10}^{\mathrm{-27}}\right]\right)$

$\mathrm{\alpha }≔1014.446097$

$\mathrm{He}≔\mathrm{Distribution}\left(\mathrm{PDF}=\mathrm{unapply}\left(g\&comma;x\right)\right)\&colon;$

$\mathrm{XHe}≔\mathrm{RandomVariable}\left(\mathrm{He}\right)\&colon;$

$\mathrm{Mode}\left(\mathrm{XHe}\right)$

$\left\{1434.643427\right\}$

$\mathrm{Mode}\left(\mathrm{Maxwell}\left(\mathrm{\alpha }\right)\right)$

$1434.643428$

$A≔\mathrm{Sample}\left(\mathrm{XHe}\&comma;{10}^5\&comma;\mathrm{method}=\left[\mathrm{envelope}\&comma;\mathrm{range}=0..5000\right]\right)\&colon;$

$\mathrm{Mode}\left(A\right)$

$1479.23422185382$

$P≔\mathrm{DensityPlot}\left(\mathrm{XHe}\&comma;\mathrm{range}=0..3500\&comma;\mathrm{thickness}=3\&comma;\mathrm{color}=\mathrm{red}\right)\&colon;$

$Q≔\mathrm{Histogram}\left(A\&comma;\mathrm{range}=0..3500\right)\&colon;$

$\mathrm{plots}\left[\mathrm{display}\right]\left(P\&comma;Q\right)$

Stuart, Alan, and Ord, Keith. Kendall's Advanced Theory of Statistics. 6th ed. London: Edward Arnold, 1998. Vol. 1: Distribution Theory.

The Statistics[Distribution] command was updated in Maple 16.