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Statistics

  

CumulativeDistributionFunction

  

compute the cumulative distribution function

 

Calling Sequence

Parameters

Description

Computation

Options

Examples

References

Calling Sequence

CumulativeDistributionFunction(X, t, options)

CDF(X, t, options)

Parameters

X

-

algebraic; random variable or distribution

t

-

algebraic; point

options

-

(optional) equation(s) of the form numeric=value or inert=value; specifies options for computing the cumulative distribution function of a random variable

Description

• 

The CumulativeDistributionFunction function computes the cumulative distribution 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]).

• 

The inverse function of the CDF is the Quantile.

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 cumulative distribution function is computed using exact arithmetic. To compute the cumulative distribution function numerically, specify the numeric or numeric=true option.

• 

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

Examples

withStatistics:

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

CumulativeDistributionFunction'Β'p,q,t

&lcub;0t<0tphypergeomp&comma;1q&comma;1&plus;p&comma;t&Beta;p&comma;qpt<11otherwise

(1)

Use numeric parameters.

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

358hypergeom4&comma;3&comma;4&comma;12

(2)

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

0.773437500000000

(3)

Define new distribution.

TDistributionPDF&equals;t&rarr;1&pi;t2&plus;1&colon;

XRandomVariableT&colon;

CDFX&comma;0

12

(4)

CDFX&comma;0&comma;numeric

0.4999999999

(5)

CDFX&comma;u

12&pi;&plus;2arctanu&pi;

(6)

Use the inert option.

CDFX&comma;0&comma;inert&equals;true

&int;&infin;01&pi;_t2&plus;1&DifferentialD;_t

(7)

CDFX&comma;t&comma;inert&equals;true

&int;&infin;t1&pi;_t02&plus;1&DifferentialD;_t0

(8)

NRandomVariableNormal0&comma;1&colon;

CDFN&comma;t&comma;inert&equals;true

&int;&infin;t122&ExponentialE;12_t12&pi;&DifferentialD;_t1

(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[Quantile]

Statistics[RandomVariables]

 


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