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Statistics

  

ProbabilityDensityFunction

  

compute the probability density function

 

Calling Sequence

Parameters

Description

Computation

Options

Examples

References

Calling Sequence

ProbabilityDensityFunction(X, t, options)

PDF(X, t, options)

Parameters

X

-

algebraic; random variable or distribution

t

-

algebraic; point

options

-

(optional) equations; specify options for computing the probability density function of a random variable

Description

• 

The ProbabilityDensityFunction function computes the probability density function of the specified random variable at the specified point.

• 

The first parameter can be a distribution (see Statistics[Distribution]), a random variable, or an algebraic expression involving random variables (see Statistics[RandomVariable]).

Computation

• 

By default, all computations involving random variables are performed symbolically (see option numeric below).

• 

For more information about computation in the Statistics package, see the Statistics[Computation] help page.

Options

  

The options argument can contain one or more of the options shown below. More information for some options is available in the Statistics[RandomVariables] help page.

• 

numeric=truefalse -- By default, the probability density function is computed using exact arithmetic. To compute the probability density function numerically, specify the numeric or numeric = true option.

• 

inert=truefalse -- By default, Maple evaluates integrals, sums, derivatives and limits encountered while computing the PDF. By specifying inert or inert=true, Maple will return these unevaluated.

• 

mainbranch - returns the main branch of the distribution only.

Examples

withStatistics:

Compute the probability density function of the beta distribution with parameters p and q.

ProbabilityDensityFunction'Β'p,q,t

0t<0t1+p1t1+qΒp&comma;qt<10otherwise

(1)

Use numeric parameters.

ProbabilityDensityFunction&apos;&Beta;&apos;3&comma;5&comma;12

10564

(2)

ProbabilityDensityFunction&apos;&Beta;&apos;3&comma;5&comma;12&comma;numeric

1.640625000

(3)

Define new distribution.

TDistributionPDF&equals;t&rarr;1Pit2&plus;1&colon;

XRandomVariableT&colon;

PDFX&comma;u

1πu2+1

(4)

PDFX&comma;0

1π

(5)

CDFX&comma;u

π+2arctanu2π

(6)

Use the inert option with a new RandomVariable, Y.

YRandomVariableDistributionCDF&equals;u&rarr;1Pi&plus;2arctanu2Pi

Y_R3

(7)

PDFY&comma;t

1t2+1π

(8)

PDFY&comma;t&comma;inert

&DifferentialD;&DifferentialD;tπ+2arctant2π

(9)

References

  

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

See Also

Statistics

Statistics[Computation]

Statistics[Distributions]

Statistics[RandomVariables]