CDF Function

Returns a value from a cumulative probability distribution.

Categories:	Probability
Categories:	CAS
Note:	The QUANTILE function returns the quantile from a distribution that you specify. The QUANTILE function is the inverse of the CDF function. For more information, see QUANTILE Function.

Table of Contents

Bernoulli Distribution

Beta Distribution

Binomial Distribution

Cauchy Distribution

Chi-Square Distribution

Conway-Maxwell-Poisson Distribution

Exponential Distribution

F Distribution

Gamma Distribution

Generalized Poisson Distribution

Geometric Distribution

Hypergeometric Distribution

Laplace Distribution

Logistic Distribution

Lognormal Distribution

Negative Binomial Distribution

Normal Distribution

Normal Mixture Distribution

Wald (Inverse Gaussian) Distribution

Weibull Distribution

Example

Syntax

CDF(distribution, quantile <, parameter-1, …, parameter-k>)

Required Arguments

distribution

is a character constant, variable, or expression that identifies the distribution. Valid distributions are as follows:

Distribution	Argument
Bernoulli	`BERNOULLI`
Beta	`BETA`
Binomial	`BINOMIAL`
Cauchy	`CAUCHY`
Chi-Square	`CHISQUARE`
Conway-Maxwell-Poisson	`CONMAXPOI`
Exponential	`EXPONENTIAL`
F	`F`
Gamma	`GAMMA`
Generalized Poisson	`GENPOISSON`
Geometric	`GEOMETRIC`
Hypergeometric	`HYPERGEOMETRIC`
Laplace	`LAPLACE`
Logistic	`LOGISTIC`
Lognormal	`LOGNORMAL`
Negative binomial	`NEGBINOMIAL`
Normal	`NORMAL\|GAUSS`
Normal mixture	`NORMALMIX`
Pareto	`PARETO`
Poisson	`POISSON`
T	`T`
Tweedie	`TWEEDIE`
Uniform	`UNIFORM`
Wald (inverse Gaussian)	`WALD\|IGAUSS`
Weibull	`WEIBULL`

Note	Except for T, F, and NORMALMIX, you can minimally identify any distribution by its first four characters.

quantile

is a numeric constant, variable, or expression that specifies the value of the random variable.

Optional Argument

parameter-1, … , parameter-k

are optional constants, variables, or expressions that specify the values of shape, location, or scale parameters that are appropriate for the specific distribution.

See	Details for complete information about these parameters.

Details

The CDF function computes the left cumulative distribution function from various continuous and discrete probability distributions.

Bernoulli Distribution

Beta Distribution

Binomial Distribution

Cauchy Distribution

Chi-Square Distribution

Conway-Maxwell-Poisson Distribution

Exponential Distribution

F Distribution

Gamma Distribution

Generalized Poisson Distribution

Geometric Distribution

Hypergeometric Distribution

Laplace Distribution

Logistic Distribution

Lognormal Distribution

Negative Binomial Distribution

Normal Distribution

Normal Mixture Distribution

Wald (Inverse Gaussian) Distribution

Weibull Distribution

Bernoulli Distribution

CDF('BERNOULLI', x, p)

Arguments

x

is a numeric constant, variable, or expression that specifies a random variable.

p

is a numeric constant, variable, or expression that specifies a probability of success.

Range

0 ≤ p ≤ 1

Details

The CDF function for the Bernoulli distribution returns the probability that an observation from a Bernoulli distribution, with probability of success equal to p, is less than or equal to x.

multiline equation line 1 upper C upper D upper F left-parenthesis prime upper B upper E upper R upper N Superscript prime Baseline comma x comma p right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column 0 2nd Column x less-than 0 2nd Row 1st Column 1 minus p 2nd Column 0 less-than-or-equal-to x less-than 1 3rd Row 1st Column 1 2nd Column x greater-than-or-equal-to 1 EndLayout

Note: There are no location or scale parameters for this distribution.

Beta Distribution

CDF('BETA', x, a, b <, l, r>)

Arguments

x

is a numeric constant, variable, or expression that specifies a random variable.

a

is a numeric constant, variable, or expression that specifies a shape parameter.

Range

a > 0

b

is a numeric constant, variable, or expression that specifies a shape parameter.

Range

b > 0

l

is a numeric constant, variable, or expression that specifies the left location parameter.

Default

r

is a numeric constant, variable, or expression that specifies the right location parameter.

Default	1
Range	r > l

Details

The CDF function for the beta distribution returns the probability that an observation from a beta distribution, with shape parameters a and b, is less than or equal to v. The following equation describes the CDF function of the beta distribution:

StartLayout 1st Row 1st Column upper C upper D upper F left-parenthesis prime upper B upper E upper T upper A Superscript prime Baseline comma x comma a comma b comma l comma r right-parenthesis 2nd Column equals StartLayout Enlarged left-brace 1st Row 1st Column 0 2nd Column x less-than-or-equal-to l 2nd Row 1st Column StartFraction 1 Over beta left-parenthesis a comma b right-parenthesis EndFraction integral Subscript l Superscript x Baseline StartFraction left-parenthesis v minus l right-parenthesis Superscript a minus 1 Baseline left-parenthesis r minus v right-parenthesis Superscript b minus 1 Baseline Over left-parenthesis r minus l right-parenthesis Superscript a plus b minus 1 Baseline EndFraction d v 2nd Column l less-than x less-than-or-equal-to r 3rd Row 1st Column 1 2nd Column x greater-than r EndLayout EndLayout

The following relationship applies to the preceding equation:

multiline equation line 1 beta left-parenthesis a comma b right-parenthesis equals StartFraction upper Gamma left-parenthesis a right-parenthesis upper Gamma left-parenthesis b right-parenthesis Over upper Gamma left-parenthesis a plus b right-parenthesis EndFraction

The following relationship applies to the preceding equation:

multiline equation line 1 upper Gamma left-parenthesis a right-parenthesis equals integral Subscript 0 Superscript infinity Baseline x Superscript a minus 1 Baseline e Superscript negative x Baseline d x

Binomial Distribution

CDF('BINOMIAL', m, p, n)

Arguments

m

is an integer random variable that counts the number of successes.

Range

m = 0, 1, ...

p

is a numeric constant, variable, or expression that specifies a probability of success parameter.

Range

0≤p ≤1

n

is a numeric constant, variable, or expression that specifies an integer parameter that counts the number of independent Bernoulli trials.

Range

n = 0, 1, ...

Details

The CDF function for the binomial distribution returns the probability that an observation from a binomial distribution, with parameters p and n, is less than or equal to m.

multiline equation line 1 upper C upper D upper F left-parenthesis prime upper B upper I upper N upper O upper M Superscript prime Baseline comma m comma p comma n right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column 0 2nd Column m less-than 0 2nd Row 1st Column sigma-summation Underscript j equals 0 Overscript m Endscripts StartBinomialOrMatrix n Choose j EndBinomialOrMatrix p Superscript j Baseline left-parenthesis 1 minus p right-parenthesis Superscript n minus j Baseline 2nd Column 0 less-than-or-equal-to m less-than-or-equal-to n 3rd Row 1st Column 1 2nd Column m greater-than n EndLayout

Note: There are no location or scale parameters for the binomial distribution.

Cauchy Distribution

CDF('CAUCHY', x <,θ> <, λ>)

Arguments

x

is a numeric constant, variable, or expression that specifies a random variable.

θ

is a numeric constant, variable, or expression that specifies a location parameter.

Default

λ

is a numeric constant, variable, or expression that specifies a scale parameter.

Default	1
Range	λ > 0

Details

The CDF function for the Cauchy distribution returns the probability that an observation from a Cauchy distribution, with the location parameter θ and the scale parameter λ, is less than or equal to x.

multiline equation line 1 upper C upper D upper F left-parenthesis prime upper C upper A upper U upper C upper H upper Y Superscript prime Baseline comma x comma theta comma lamda right-parenthesis equals one-half plus StartFraction 1 Over pi EndFraction t a n Superscript negative 1 Baseline left-parenthesis StartFraction x minus theta Over lamda EndFraction right-parenthesis

Chi-Square Distribution

CDF('CHISQUARE', x, df <, nc>)

Arguments

x

is a numeric constant, variable, or expression that specifies a random variable.

df

is a numeric constant, variable, or expression that specifies a degrees of freedom parameter.

Range

df > 0

nc

is a numeric constant, variable, or expression that specifies an optional noncentrality parameter.

Range

nc ≥ 0

Details

The CDF function for the chi-square distribution returns the probability that an observation from a chi-square distribution, with df degrees of freedom and the noncentrality parameter nc, is less than or equal to x. This function accepts non-integer degrees of freedom. If nc is omitted or equal to zero, the value returned is from the central chi-square distribution. In the following equation, let

and let

. The following equation describes the CDF function of the chi-square distribution:

multiline equation line 1 upper C upper D upper F left-parenthesis prime upper C upper H upper I upper S upper Q Superscript prime Baseline comma x comma nu comma lamda right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column 0 2nd Column x less-than 0 2nd Row 1st Column sigma-summation Underscript j equals 0 Overscript infinity Endscripts e Superscript minus StartFraction lamda Over 2 EndFraction Baseline times StartFraction left-parenthesis StartFraction lamda Over 2 EndFraction right-parenthesis Superscript j Baseline Over j factorial EndFraction upper P Subscript c Baseline left-parenthesis x comma nu plus 2 j right-parenthesis 2nd Column x greater-than-or-equal-to 0 EndLayout

In the equation, P_c(.,.) denotes the probability from the central chi-square distribution:

multiline equation line 1 upper P Subscript c Baseline left-parenthesis x comma a right-parenthesis equals upper P Subscript g Baseline left-parenthesis StartFraction x Over 2 EndFraction comma StartFraction a Over 2 EndFraction right-parenthesis

In the equation, P_g(y,b) is the probability from the gamma distribution given by the equation:

multiline equation line 1 upper P Subscript g Baseline left-parenthesis y comma b right-parenthesis equals StartFraction 1 Over upper Gamma left-parenthesis b right-parenthesis EndFraction integral Subscript 0 Superscript y Baseline e Superscript negative v Baseline v Superscript b minus 1 Baseline d v

Conway-Maxwell-Poisson Distribution

CDF('CONMAXPOI', y, λ,ν)

Arguments

y

is a numeric constant, variable, or expression that specifies a nonnegative integer that represents counts data.

λ

is similar to the mean, as in the Poisson distribution.

ν

is a numeric constant, variable, or expression that specifies a dispersion parameter.

Details

The CDF function returns cumulative probability from 0 to y. For more information, see “Conway-Maxwell-Poisson” distribution in the PDF function .

Exponential Distribution

CDF('EXPONENTIAL', x <,λ>)

Arguments

x

is a numeric constant, variable, or expression that specifies a random variable.

λ

is a numeric constant, variable, or expression that specifies a scale parameter.

Default	1
Range	λ > 0

Details

The CDF function for the exponential distribution returns the probability that an observation from an exponential distribution, with the scale parameter λ, is less than or equal to x.

multiline equation line 1 upper C upper D upper F left-parenthesis prime upper E upper X upper P upper O Superscript prime Baseline comma x comma lamda right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column 0 2nd Column x less-than 0 2nd Row 1st Column 1 minus e Superscript minus StartFraction x Over lamda EndFraction Baseline 2nd Column x greater-than-or-equal-to 0 EndLayout

F Distribution

CDF('F', x, ndf, ddf <, nc>)

Arguments

x

is a numeric constant, variable, or expression that specifies a random variable.

ndf

is a numeric constant, variable, or expression that specifies a numerator degrees of freedom parameter.

Range

ndf > 0

ddf

is a numeric constant, variable, or expression that specifies a denominator degrees of freedom parameter.

Range

ddf > 0

nc

is a numeric constant, variable, or expression that specifies a noncentrality parameter.

Range

nc ≥ 0

Details

The CDF function for the F distribution returns the probability that an observation from an F distribution, with ndf numerator degrees of freedom, ddf denominator degrees of freedom, and the noncentrality parameter nc, is less than or equal to x. This function accepts noninteger degrees of freedom for ndf and ddf. If nc is omitted or equal to zero, the value returned is from a central F distribution. In the following equation, let

, let

, and let

. The following equation describes the CDF function of the F distribution:

StartLayout 1st Row 1st Column Blank 2nd Column upper C upper D upper F left-parenthesis prime upper F Superscript prime Baseline comma x comma v 1 comma v 2 comma lamda right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column 0 2nd Column x less-than 0 2nd Row 1st Column sigma-summation Underscript j equals 0 Overscript infinity Endscripts e Superscript minus StartFraction lamda Over 2 EndFraction times Baseline StartFraction left-parenthesis StartFraction lamda Over 2 EndFraction right-parenthesis Superscript j Baseline Over j factorial EndFraction upper P Subscript upper F Baseline left-parenthesis x comma v 1 plus 2 j comma v 2 right-parenthesis 2nd Column x greater-than-or-equal-to 0 EndLayout 2nd Row 1st Column Blank EndLayout

In the equation, P_f(f,u₁,u₂) is the probability from the central F distribution with

StartLayout 1st Row 1st Column Blank 2nd Column upper P Subscript upper F Baseline left-parenthesis x comma u 1 comma u 2 right-parenthesis equals upper P Subscript upper B Baseline left-parenthesis StartFraction u 1 x Over u 1 x plus u 2 EndFraction comma StartFraction u 1 Over 2 EndFraction comma StartFraction u 2 Over 2 EndFraction right-parenthesis EndLayout

and P_B(x,a,b) is the probability from the standard beta distribution.

Note: There are no location or scale parameters for the F distribution.

Gamma Distribution

CDF('GAMMA', x, a <, λ>)

Arguments

x

is a numeric constant, variable, or expression that specifies a random variable.

a

is a numeric constant, variable, or expression that specifies a shape parameter.

Range

a > 0

λ

is a numeric constant, variable, or expression that specifies a scale parameter.

Default	1
Range	λ > 0

Details

The CDF function for the gamma distribution returns the probability that an observation from a gamma distribution, with the shape parameter a and the scale parameter λ, is less than or equal to x.

multiline equation line 1 upper C upper D upper F left-parenthesis prime upper G upper A upper M upper M upper A Superscript prime Baseline comma x comma a comma lamda right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column 0 2nd Column x less-than 0 2nd Row 1st Column StartFraction 1 Over lamda Superscript a Baseline upper Gamma left-parenthesis a right-parenthesis EndFraction integral Subscript 0 Superscript x Baseline v Superscript a minus 1 Baseline e Superscript minus StartFraction v Over lamda EndFraction Baseline d v 2nd Column x greater-than-or-equal-to 0 EndLayout

Generalized Poisson Distribution

CDF(‘GENPOISSON’, x, θ, η)

Arguments

x

is a numeric constant, variable, or expression that specifies an integer random variable.

θ

is a numeric constant, variable, or expression that specifies a shape parameter.

Range

≤5 and >0

η

is a numeric constant, variable, or expression that specifies a shape parameter.

Range	≥0 and <0.95
Tip	When η =0, the distribution is the Poisson distribution with a mean and variance of θ. When η>0, the mean is and the variance is .

Details

The probability mass function for the generalized Poisson distribution follows:

f left-parenthesis x semicolon theta comma eta right-parenthesis equals theta left-parenthesis theta plus eta x right-parenthesis Superscript x minus 1 Baseline e Superscript negative theta minus eta x Baseline slash x factorial comma x equals 0 comma 1 comma 2 comma period period period comma theta greater-than 0 comma 0 less-than-or-equal-to eta less-than 1

If η =0, then the generalized Poisson distribution becomes the standard Poisson distribution with the shape parameter θ.

Geometric Distribution

CDF('GEOMETRIC', m, p)

Arguments

m

is a numeric random variable that specifies the number of failures.

Range

m = 0, 1, ...

p

is a numeric constant, variable, or expression that specifies a probability of success.

Range

0 ≤ p ≤ 1

Details

The CDF function for the geometric distribution returns the probability that an observation from a geometric distribution, with the parameter p, is less than or equal to m.

multiline equation line 1 upper C upper D upper F left-parenthesis prime upper G upper E upper O upper M Superscript prime Baseline comma m comma p right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column 0 2nd Column m less-than 0 2nd Row 1st Column 1 minus left-parenthesis 1 minus p right-parenthesis Superscript left-parenthesis m plus 1 right-parenthesis Baseline 2nd Column m greater-than-or-equal-to 0 EndLayout

Note: There are no location or scale parameters for this distribution.

Hypergeometric Distribution

CDF('HYPER', x, N, R, n <, o>)

Arguments

x

is a numeric constant, variable, or expression that specifies an integer random variable.

N

is a numeric constant, variable, or expression that specifies an integer population size parameter.

Range

N = 1, 2, ...

R

is a numeric constant, variable, or expression that specifies an integer number of items in the category of interest.

Range

R = 0, 1, ..., N

n

is a numeric constant, variable, or expression that specifies an integer sample size parameter.

Range

n = 1, 2, ..., N

o

is a numeric constant, variable, or expression that specifies an optional numeric odds ratio parameter.

Range

o > 0

Details

The CDF function for the hypergeometric distribution returns the probability that an observation from an extended hypergeometric distribution, with population size N, number of items R, sample size n, and odds ratio o, is less than or equal to x. If o is omitted or equal to 1, the value returned is from the usual hypergeometric distribution.

StartLayout 1st Row 1st Column upper C upper D upper F 2nd Column left-parenthesis prime upper H upper Y upper P upper E upper R Superscript prime Baseline comma x comma upper N comma upper R comma n comma o right-parenthesis equals 2nd Row 1st Column Blank 2nd Column StartLayout Enlarged left-brace 1st Row 1st Column 0 2nd Column x less-than m a x left-parenthesis 0 comma upper R plus n minus upper N right-parenthesis 2nd Row 1st Column StartFraction sigma-summation Underscript i equals 0 Overscript x Endscripts StartBinomialOrMatrix upper R Choose i EndBinomialOrMatrix StartBinomialOrMatrix upper N minus upper R Choose n minus i EndBinomialOrMatrix o Superscript i Baseline Over sigma-summation Underscript j equals m a x left-parenthesis 0 comma upper R plus n minus upper N right-parenthesis Overscript m i n left-parenthesis upper R comma n right-parenthesis Endscripts StartBinomialOrMatrix upper R Choose j EndBinomialOrMatrix StartBinomialOrMatrix upper N minus upper R Choose n minus j EndBinomialOrMatrix o Superscript j Baseline EndFraction 2nd Column m a x left-parenthesis 0 comma upper R plus n minus upper N right-parenthesis less-than-or-equal-to x less-than-or-equal-to m i n left-parenthesis upper R comma n right-parenthesis 3rd Row 1st Column 1 2nd Column x greater-than m i n left-parenthesis upper R comma n right-parenthesis EndLayout EndLayout

Laplace Distribution

CDF('LAPLACE', x <,θ, λ>)

Arguments

x

is a numeric constant, variable, or expression that specifies a random variable.

θ

is a numeric constant, variable, or expression that specifies a location parameter.

Default

λ

is a numeric constant, variable, or expression that specifies a scale parameter.

Default	1
Range	λ > 0

Details

The CDF function for the Laplace distribution returns the probability that an observation from the Laplace distribution, with the location parameter θ and the scale parameter λ, is less than or equal to x.

multiline equation line 1 upper C upper D upper F left-parenthesis prime upper L upper A upper P upper L upper A upper C upper E Superscript prime Baseline comma x comma theta comma lamda right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column one-half e Superscript StartFraction left-parenthesis x minus theta right-parenthesis Over lamda EndFraction Baseline 2nd Column x less-than theta 2nd Row 1st Column 1 minus one-half e Superscript left-parenthesis minus StartFraction left-parenthesis x minus theta right-parenthesis Over lamda EndFraction right-parenthesis Baseline 2nd Column x greater-than-or-equal-to theta EndLayout

Logistic Distribution

CDF('LOGISTIC', x <, θ, λ>)

Arguments

x

is a numeric constant, variable, or expression that specifies a random variable.

θ

is a numeric constant, variable, or expression that specifies a location parameter.

Default

λ

is a numeric constant, variable, or expression that specifies a scale parameter.

Default	1
Range	λ > 0

Details

The CDF function for the Logistic distribution returns the probability that an observation from a Logistic distribution, with the location parameter θ and the scale parameter λ, is less than or equal to x.

multiline equation line 1 upper C upper D upper F left-parenthesis prime upper L upper O upper G upper I upper S upper T upper I upper C Superscript prime Baseline comma x comma theta comma lamda right-parenthesis equals StartFraction 1 Over 1 plus e Superscript left-parenthesis minus StartFraction x minus theta Over lamda EndFraction right-parenthesis Baseline EndFraction

Lognormal Distribution

CDF('LOGNORMAL', x <, θ, λ>)

Arguments

x

is a numeric constant, variable, or expression that specifies a random variable.

θ

is a numeric constant, variable, or expression that specifies a log scale parameter. e(θ) is a scale parameter.

Default

λ

is a numeric constant, variable, or expression that specifies a shape parameter.

Default	1
Range	λ > 0

Details

The CDF function for the lognormal distribution returns the probability that an observation from a lognormal distribution, with the log scale parameter θ and the shape parameter λ, is less than or equal to x.

multiline equation line 1 upper C upper D upper F left-parenthesis prime upper L upper O upper G upper N Superscript prime Baseline comma x comma theta comma lamda right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column 0 2nd Column x less-than-or-equal-to 0 2nd Row 1st Column StartFraction 1 Over lamda RootIndex StartRoot 2 pi EndRoot EndFraction integral Subscript negative infinity Superscript l o g left-parenthesis x right-parenthesis Baseline e Superscript left-parenthesis minus StartFraction left-parenthesis v minus theta right-parenthesis squared Over 2 lamda squared EndFraction right-parenthesis Baseline times d v 2nd Column x greater-than 0 EndLayout

Negative Binomial Distribution

CDF('NEGBINOMIAL', m, p, n)

Arguments

m

is a numeric constant, variable, or expression that specifies a positive integer random variable that counts the number of failures.

Range

m = 0, 1, ...

p

is a numeric constant, variable, or expression that specifies a probability of success.

Range

0 ≤ p ≤ 1

n

is a numeric constant, variable, or expression that specifies a value that counts the number of successes.

Range

n > 0

Details

The CDF function for the negative binomial distribution returns the probability that an observation from a negative binomial distribution, with the probability of success p and the number of successes n, is less than or equal to m.

multiline equation line 1 upper C upper D upper F left-parenthesis prime upper N upper E upper G upper B Superscript prime Baseline comma m comma p comma n right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column 0 2nd Column m less-than 0 2nd Row 1st Column p Superscript n Baseline normal upper Sigma Underscript j equals 0 Overscript m Endscripts StartBinomialOrMatrix n plus j minus 1 Choose n minus 1 EndBinomialOrMatrix left-parenthesis 1 minus p right-parenthesis Superscript j Baseline 2nd Column m greater-than-or-equal-to 0 EndLayout

Note: There are no location or scale parameters for the negative binomial distribution.

Normal Distribution

CDF('NORMAL', x <, θ, λ>)

Arguments

x

is a numeric constant, variable, or expression that specifies a random variable.

θ

is a numeric constant, variable, or expression that specifies a location parameter.

Default

λ

is a numeric constant, variable, or expression that specifies a scale parameter.

Default	1
Range	λ > 0

Details

The CDF function for the Normal distribution returns the probability that an observation from the Normal distribution, with the location parameter θ and the scale parameter λ, is less than or equal to x.

multiline equation line 1 upper C upper D upper F left-parenthesis prime upper N upper O upper R upper M upper A upper L Superscript prime Baseline comma x comma theta comma lamda right-parenthesis equals StartFraction 1 Over lamda RootIndex StartRoot 2 pi EndRoot EndFraction integral Subscript negative infinity Superscript x Baseline e Superscript left-parenthesis minus StartFraction left-parenthesis v minus theta right-parenthesis squared Over 2 lamda squared EndFraction right-parenthesis Baseline times d v

Normal Mixture Distribution

CDF('NORMALMIX', x, n, p, m, s)

Arguments

x

is a numeric constant, variable, or expression that specifies a random variable.

n

is a numeric constant, variable, or expression that specifies the number of mixtures.

Range

n = 1, 2, ...

p

is a numeric constant, variable, or expression that specifies the n proportions, , where .

Range

p = 0, 1, ...

m

is a numeric constant, variable, or expression that specifies the n means .

s

is a numeric constant, variable, or expression that specifies the n standard deviations .

Range

s > 0

Details

The CDF function for the Normal Mixture distribution returns the probability that an observation from a mixture of normal distribution is less than or equal to x.

multiline equation line 1 upper C upper D upper F left-parenthesis prime upper N upper O upper R upper M upper A upper L upper M upper I upper X prime comma x comma n comma p comma m comma s right-parenthesis equals normal upper Sigma Underscript i equals 1 Overscript i equals n Endscripts p Subscript i Baseline upper C upper D upper F left-parenthesis prime upper N upper O upper R upper M upper A upper L prime comma x comma m Subscript i Baseline comma s Subscript i Baseline right-parenthesis

Weights for the Normal Mixture distribution must be nonnegative. If the sum of the weights does not equal 1, then the weights are treated as relative weights and adjusted so that the sum equals 1.

Note: There are no location or scale parameters for the Normal Mixture distribution.

Pareto Distribution

CDF('PARETO', x, a <, k>)

Arguments

x

is a numeric constant, variable, or expression that specifies a random variable.

a

is a numeric constant, variable, or expression that specifies a shape parameter.

Range

a > 0

k

is a numeric constant, variable, or expression that specifies a scale parameter.

Default	1
Range	k > 0

Details

The CDF function for the Pareto distribution returns the probability that an observation from a Pareto distribution, with the shape parameter a and the scale parameter k, is less than or equal to x.

multiline equation line 1 upper C upper D upper F left-parenthesis prime upper P upper A upper R upper E upper T upper O Superscript prime Baseline comma x comma a comma k right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column 0 2nd Column x less-than k 2nd Row 1st Column 1 minus left-parenthesis StartFraction k Over x EndFraction right-parenthesis Superscript a Baseline 2nd Column x greater-than-or-equal-to k EndLayout

Poisson Distribution

CDF('POISSON', n, m)

Arguments

n

is a numeric constant, variable, or expression that specifies an integer random variable.

Range

n = 0, 1, ...

m

is a numeric constant, variable, or expression that specifies a mean parameter.

Range

m > 0

Details

The CDF function for the Poisson distribution returns the probability that an observation from a Poisson distribution, with mean m, is less than or equal to n.

multiline equation line 1 upper C upper D upper F left-parenthesis prime upper P upper O upper I upper S upper S upper O upper N prime comma n comma m right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column 0 2nd Column n less-than 0 2nd Row 1st Column normal upper Sigma Underscript i equals 0 Overscript n Endscripts e Superscript negative m Baseline StartFraction m Superscript i Baseline Over i factorial EndFraction 2nd Column n greater-than-or-equal-to 0 EndLayout

Note: There are no location or scale parameters for the Poisson distribution.

T Distribution

CDF('T', t, df <, nc>)

Arguments

t

is a numeric constant, variable, or expression that specifies a random variable.

df

is a numeric constant, variable, or expression that specifies the degrees of freedom.

Range

df > 0

nc

is a numeric constant, variable, or expression that specifies an optional noncentrality parameter.

Details

The CDF function for the T distribution returns the probability that an observation from a T distribution, with degrees of freedom df and the noncentrality parameter nc, is less than or equal to x. This function accepts noninteger degrees of freedom. If nc is omitted or equal to zero, the value returned is from the central T distribution. In the following equation, let

and let

multiline equation line 1 upper C upper D upper F left-parenthesis prime upper T Superscript prime Baseline comma t comma nu comma delta right-parenthesis equals StartStartFraction 1 OverOver 2 Superscript left-parenthesis nu slash 2 minus 1 right-parenthesis Baseline upper Gamma left-parenthesis StartFraction nu Over 2 EndFraction right-parenthesis EndEndFraction integral Subscript 0 Superscript infinity Baseline x Superscript nu minus 1 Baseline e Superscript minus one-half x squared Baseline StartFraction 1 Over RootIndex StartRoot 2 pi EndRoot EndFraction integral Subscript negative infinity Superscript StartFraction t x Over RootIndex StartRoot nu EndRoot EndFraction Baseline e Superscript minus one-half left-parenthesis u minus delta right-parenthesis squared Baseline d u d x

Note: There are no location or scale parameters for the T distribution.

Tweedie Distribution

CDF (‘TWEEDIE’, y, p <, µ, φ>)

Arguments

y

is a numeric constant, variable, or expression that specifies a random variable.

Range	y ≥0
Notes	This argument is required.
Notes	When p>1, y is numeric. When p=1, y is an integer.

p

is a numeric constant, variable, or expression that specifies the power parameter.

Range	p ≥1
Note	This argument is required.

µ

is a numeric constant, variable, or expression that specifies the mean parameter.

Default	1
Range	µ>0

φ

is a numeric constant, variable, or expression that specifies the dispersion parameter.

Default	1
Range	φ>0

Details

The CDF function for the Tweedie distribution returns an exponential dispersion model with variance and mean related by the equation variance

integral Subscript 0 Superscript y Baseline StartFraction 1 Over y EndFraction ModifyingAbove normal upper Sigma With infinity Underscript j equals 1 Endscripts left-parenthesis StartFraction y Superscript minus j alpha Baseline left-parenthesis p minus 1 right-parenthesis Superscript j a Baseline Over phi Superscript j left-parenthesis 1 minus alpha right-parenthesis left-parenthesis 2 minus p right-parenthesis Super Superscript j Superscript j factorial normal upper Gamma left-parenthesis minus j alpha right-parenthesis Baseline EndFraction right-parenthesis e Superscript left-parenthesis StartFraction 1 Over phi EndFraction left-parenthesis y StartFraction mu Super Superscript 1 minus p Superscript minus 1 Over 1 minus p EndFraction minus StartFraction mu Super Superscript 2 minus p Superscript minus 1 Over 2 minus p EndFraction right-parenthesis right-parenthesis Baseline d y

The following relationship applies to the preceding equation:

alpha equals StartFraction 2 minus p Over 1 minus p EndFraction

Note: The accuracy of computed Tweedie probabilities is highly dependent on the location in parameter space. Ten digits of accuracy are usually available except when p is near 2 or phi is near 0. In that case, the accuracy might be as low as six digits.

Uniform Distribution

CDF('UNIFORM', x <, l, r>)

Arguments

x

is a numeric constant, variable, or expression that specifies a random variable.

l

is a numeric constant, variable, or expression that specifies the left location parameter.

Default

r

is a numeric constant, variable, or expression that specifies the right location parameter.

Default	1
Range	r > l

Details

The CDF function for the uniform distribution returns the probability that an observation from a uniform distribution, with the left location parameter l and the right location parameter r, is less than or equal to x.

multiline equation line 1 upper C upper D upper F left-parenthesis prime upper U upper N upper I upper F upper O upper R upper M Superscript prime Baseline comma x comma l comma r right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column 0 2nd Column x less-than l 2nd Row 1st Column StartFraction x minus l Over r minus l EndFraction 2nd Column l less-than-or-equal-to x less-than r 3rd Row 1st Column 1 2nd Column x greater-than-or-equal-to r EndLayout

Note: The default values for l and r are 0 and 1, respectively.

Wald (Inverse Gaussian) Distribution

CDF('WALD', x, λ <, µ>)

CDF('IGAUSS', x, λ <, µ>)

Arguments

x

is a numeric constant, variable, or expression that specifies a random variable.

λ

is a numeric constant, variable, or expression that specifies a shape parameter.

Range

λ > 0

µ

is a numeric constant, variable, or expression that specifies the mean parameter.

Default	1
Range	µ > 0

Details

The CDF function for the Wald distribution returns the probability that an observation from a Wald distribution, with the shape parameter λ, is less than or equal to x.

upper F x left-parenthesis x right-parenthesis equals normal upper Phi left-brace StartRoot StartFraction lamda Over x EndFraction EndRoot left-parenthesis StartFraction x Over mu EndFraction minus 1 right-parenthesis right-brace plus e Superscript 2 lamda slash mu Baseline normal upper Phi left-brace minus StartRoot StartFraction lamda Over x EndFraction EndRoot left-parenthesis StartFraction x Over mu EndFraction plus 1 right-parenthesis right-brace

In the equation,

(.) is the standard normal cumulative distribution function. When x≤0, CDF is 0.

Weibull Distribution

CDF('WEIBULL', x, a <, λ>)

Arguments

x

is a numeric constant, variable, or expression that specifies a random variable.

a

is a numeric constant, variable, or expression that specifies a shape parameter.

Range

a > 0

λ

is a numeric constant, variable, or expression that specifies a scale parameter.

Default	1
Range	λ > 0

Details

The CDF function for the Weibull distribution returns the probability that an observation from a Weibull distribution, with the shape parameter a and the scale parameter λ, is less than or equal to x.

multiline equation line 1 upper C upper D upper F left-parenthesis prime upper W upper E upper I upper B upper U upper L upper L Superscript prime Baseline comma x comma a comma lamda right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column 0 2nd Column x less-than 0 2nd Row 1st Column 1 minus e Superscript minus left-parenthesis StartFraction x Over lamda EndFraction right-parenthesis Super Superscript a Baseline 2nd Column x greater-than-or-equal-to 0 EndLayout

Example

The following SAS statements produce these results:

SAS Statement	Result
y=cdf('BERN', 0, .25);	0.75
y=cdf('BETA', 0.2, 3, 4);	0.09888
y=cdf('BINOM', 4, .5, 10);	0.37695
y=cdf('CAUCHY', 2);	0.85242
y=cdf('CHISQ', 11.264, 11);	0.57858
y=cdf('CONMAXPOI', 5, 2.3, .4);	0.2445411535
y=cdf('EXPO', 1);	0.63212
y=cdf('F', 3.32, 2, 3);	0.82639
y=cdf('GAMMA', 1, 3);	0.080301
y=cdf('GENPOISSON', 9, 1, .7);	0.906162963
y=cdf('HYPER', 2, 200, 50, 10);	0.52367
y=cdf('LAPLACE', 1);	0.81606
y=cdf('LOGISTIC', 1);	0.73106
y=cdf('LOGNORMAL', 1);	0.5
y=cdf('NEGB', 1, .5, 2);	0.5
y=cdf('NORMAL', 1.96);	0.97500
y=cdf('NORMALMIX',2.3, 3,.33, .33, .34, .5, 1.5, 2.5, .79, 1.6, 4.3);	0.7181
y=cdf('PARETO', 1 ,1);	0
y=cdf('POISSON', 2, 1);	0.91970
y=cdf('T', .9, 5);	0.79531
y=cdf('TWEEDIE', .8, 5);	0.5917629164
y=cdf('UNIFORM', 0.25);	0.25
y=cdf('WALD', 1, 2);	0.62770
y=cdf('WEIBULL', 1, 2);	0.63212