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 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 numeric 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

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

PDF ('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 the probability of success.

Range

0≤p≤1

Details

The PDF function for the Bernoulli distribution returns the probability density function with the probability of success equal to p. The PDF function is evaluated at the value x.

multiline equation line 1 upper P 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 x equals 0 3rd Row 1st Column 0 2nd Column 0 less-than x less-than 1 4th Row 1st Column p 2nd Column x equals 1 5th Row 1st Column 0 2nd Column x greater-than 1 EndLayout

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

Beta Distribution

PDF ('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 PDF function for the beta distribution returns the probability density function with the shape parameters a and b. The PDF function is evaluated at the value x.

multiline equation line 1 upper P 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 equals StartLayout Enlarged left-brace 1st Row 1st Column 0 2nd Column x less-than l 2nd Row 1st Column StartFraction 1 Over beta left-parenthesis a comma b right-parenthesis EndFraction StartFraction left-parenthesis x minus l right-parenthesis Superscript a minus 1 Baseline left-parenthesis r minus x right-parenthesis Superscript b minus 1 Baseline Over left-parenthesis r minus l right-parenthesis Superscript a plus b minus 1 Baseline EndFraction 2nd Column l less-than-or-equal-to x less-than-or-equal-to r 3rd Row 1st Column 0 2nd Column x greater-than r EndLayout

Note: The quantity

StartFraction x minus l Over r minus l EndFraction

is forced to be

epsilon less-than-or-equal-to StartFraction x minus l Over r minus l EndFraction less-than-or-equal-to 1 minus 2 epsilon

Binomial Distribution

PDF ('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 the probability of success.

Range

0 ≤ p ≤ 1

n

is an integer parameter that counts the number of independent Bernoulli trials.

Range

n=0, 1, ...

Details

The PDF function for the binomial distribution returns the probability density function with the parameters p and n. The PDF function is evaluated at the value m.

multiline equation line 1 upper P 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 StartBinomialOrMatrix n Choose m EndBinomialOrMatrix p Superscript m Baseline left-parenthesis 1 minus p right-parenthesis Superscript n minus m Baseline 2nd Column 0 less-than-or-equal-to m less-than-or-equal-to n 3rd Row 1st Column 0 2nd Column m greater-than n EndLayout

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

Cauchy Distribution

PDF ('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 PDF function for the Cauchy distribution returns the probability density function with the location parameter θ and the scale parameter λ. The PDF function is evaluated at the value x.

multiline equation line 1 upper P 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 StartFraction 1 Over pi EndFraction left-parenthesis StartFraction lamda Over lamda squared plus left-parenthesis x minus theta right-parenthesis squared EndFraction right-parenthesis

Chi-Square Distribution

PDF ('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 the degrees of freedom.

Range

df > 0

nc

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

Range

nc ≥ 0

Details

The PDF function for the chi-square distribution returns the probability density function of a chi-square distribution, with df degrees of freedom and the noncentrality parameter nc. The PDF function is evaluated at the value x. This function accepts noninteger degrees of freedom. If nc is omitted or equal to zero, the value returned is from the central chi-square distribution.

multiline equation line 1 upper P 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 ModifyingAbove normal upper Sigma With infinity Underscript j equals 0 Endscripts e Superscript minus StartFraction lamda Over 2 EndFraction Baseline StartFraction left-parenthesis StartFraction lamda Over 2 EndFraction right-parenthesis Superscript j Baseline Over j factorial EndFraction 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 this equation, p_c(.,.) denotes the density from the central chi-square distribution:

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

In this equation, p_g(y,b) is the density from the gamma distribution:

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

Conway-Maxwell-Poisson Distribution

PDF('CONMAXPOI',y,λ,ν)

Arguments

y

is a numeric constant, variable, or expression that specifies a nonnegative integer representing a count.

λ

is a numeric constant, variable, or expression that specifies a location parameter, similar to the Poisson mean parameter.

ν

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

Details

The Conway-Maxwell-Poisson (CMP) distribution is a generalization of the Poisson distribution that enables you to model underdispersed and overdispersed data. The CMP distribution is defined according to this equation:

upper P left-parenthesis upper Y equals y times semicolon lamda comma nu right-parenthesis equals StartFraction 1 Over upper Z left-parenthesis lamda comma nu right-parenthesis EndFraction StartFraction lamda Superscript y Baseline Over left-parenthesis y factorial right-parenthesis Superscript nu Baseline EndFraction times y equals 0 comma 1 comma 2 comma period period period

The normalization factor is expressed by this equation:

upper Z left-parenthesis lamda comma nu right-parenthesis equals sigma-summation Underscript n equals 0 Overscript infinity Endscripts StartFraction lamda Superscript n Baseline Over left-parenthesis n factorial right-parenthesis Superscript nu Baseline EndFraction

λ and ν are nonnegative and not simultaneously zero.

The additional parameter, ν, allows for flexibility in modeling the tail behavior of the distribution. If ν=1, the ratio is equal to the rate of decay of the Poisson distribution. If ν<1, the rate of decay decreases, which enables you to model processes that have longer tails than the Poisson distribution (overdispersed data). If ν>1, the rate of decay increases in a nonlinear manner, thus shortening the tail of the distribution (underdispersed data).

There are several special cases of the Conway-Maxwell-Poisson distribution. If λ<1 and ν→∞, the Conway-Maxwell-Poisson distribution results in the Bernoulli distribution. In this case, the data can take only the values 0 and 1, which represent an extreme underdispersion. If ν=1, the Poisson distribution is recovered with its equidispersion property. When ν=0 and λ<1, the normalization factor is convergent and forms this geometric series:

upper Z left-parenthesis lamda comma 0 right-parenthesis equals StartFraction 1 Over 1 minus lamda EndFraction

The probability density function is represented by this equation:

upper P left-parenthesis upper Y equals y times semicolon lamda comma nu equals 0 right-parenthesis equals left-parenthesis 1 minus lamda right-parenthesis lamda Superscript y

The geometric distribution represents a case of severe overdispersion.

Mean, Variance, and Dispersion for the Conway-Maxwell-Poisson Model

The mean and variance of the Conway-Maxwell-Poisson distribution are defined by these equations:

upper E left-bracket upper Y right-bracket equals StartFraction partial-differential 1 n times upper Z Over partial-differential 1 n times lamda EndFraction

upper V left-bracket upper Y right-bracket equals StartFraction partial-differential squared 1 n times upper Z Over partial-differential squared 1 n times lamda EndFraction

The Conway-Maxwell-Poisson distribution does not have closed-form expressions for its moments in terms of parameters λ and ν. However, the moments can be approximated. (For more information about the Conway-Maxwell-Poisson distribution and discrete data, see the References section that is located at the end of this function.) Use asymptotic expressions for Z to derive E(Y) and V(Y) as these equations show:

upper E left-bracket upper Y right-bracket almost-equals lamda Superscript 1 slash nu Baseline plus StartFraction 1 Over 2 nu EndFraction minus one-half

upper V left-bracket upper Y right-bracket almost-equals StartFraction 1 Over nu EndFraction lamda Superscript 1 slash nu

In the Conway-Maxwell-Poisson model, the summation of infinite series is evaluated using a logarithmic expansion. (For more information about the Conway-Maxwell-Poisson distribution and discrete data, see the References section that is located at the end of this function.) The mean and variance are calculated as follows for the Conway-Maxwell-Poisson model:

upper E left-parenthesis upper Y right-parenthesis equals StartFraction 1 Over upper Z left-parenthesis lamda comma nu right-parenthesis EndFraction sigma-summation Underscript j equals 0 Overscript infinity Endscripts StartFraction j Superscript lamda Super Superscript j Superscript Baseline Over left-parenthesis j factorial right-parenthesis Superscript nu Baseline EndFraction

upper V left-parenthesis upper Y right-parenthesis equals StartFraction 1 Over upper Z left-parenthesis lamda comma nu right-parenthesis EndFraction sigma-summation Underscript j equals 0 Overscript infinity Endscripts StartFraction j squared lamda Superscript j Baseline Over left-parenthesis j factorial right-parenthesis Superscript nu Baseline EndFraction minus upper E left-parenthesis upper Y right-parenthesis squared

The dispersion is defined as follows:

upper D left-parenthesis upper Y right-parenthesis equals StartFraction upper V left-parenthesis upper Y right-parenthesis Over upper E left-parenthesis upper Y right-parenthesis EndFraction

Exponential Distribution

PDF ('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 PDF function for the exponential distribution returns the probability density function of an exponential distribution, with the scale parameter λ. The PDF function is evaluated at the value x.

multiline equation line 1 upper P 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 StartFraction 1 Over lamda EndFraction e x p left-parenthesis minus StartFraction x Over lamda EndFraction right-parenthesis 2nd Column x greater-than-or-equal-to 0 EndLayout

F Distribution

PDF ('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 the numerator degrees of freedom.

Range

ndf> 0

ddf

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

Range

ddf > 0

nc

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

Range

nc ≥ 0

Details

The PDF function for the F distribution returns the probability density function of an F distribution, with ndf numerator degrees of freedom, ddf denominator degrees of freedom, and the noncentrality parameter nc. The PDF function is evaluated at the value x. This PDF 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

. This equation describes the PDF function for the F distribution:

multiline equation line 1 upper P upper D upper F left-parenthesis prime upper F Superscript prime Baseline comma x comma nu 1 comma nu 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 ModifyingAbove normal upper Sigma With infinity Underscript j equals 0 Endscripts e Superscript minus StartFraction lamda Over 2 EndFraction Baseline StartFraction left-parenthesis StartFraction lamda Over 2 EndFraction right-parenthesis Superscript j Baseline Over j factorial EndFraction p Subscript f Baseline left-parenthesis f comma nu 1 plus 2 j comma nu 2 right-parenthesis 2nd Column x greater-than-or-equal-to 0 EndLayout

In the equation, p_f(f,u₁,u₂) is the density from the central F distribution:

multiline equation line 1 p Subscript f Baseline left-parenthesis f comma u 1 comma u 2 right-parenthesis equals p Subscript upper B Baseline left-parenthesis StartFraction u 1 f Over u f plus u 2 EndFraction comma StartFraction u 1 Over 2 EndFraction comma StartFraction u 2 Over 2 EndFraction right-parenthesis StartFraction u 1 u 2 Over left-parenthesis u 2 plus u 1 f right-parenthesis squared EndFraction

In the equation, p_B(x,a,b) is the density from the standard beta distribution.

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

Gamma Distribution

PDF ('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 PDF function for the gamma distribution returns the probability density function of a gamma distribution, with the shape parameter a and the scale parameter λ. The PDF function is evaluated at the value x.

multiline equation line 1 upper P 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 x Superscript a minus 1 Baseline e x p left-parenthesis minus StartFraction x Over lamda EndFraction right-parenthesis 2nd Column x greater-than-or-equal-to 0 EndLayout

Generalized Poisson Distribution

PDF ('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

<10⁵ 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

Here is the probability mass function for the generalized Poisson distribution:

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

Geometric Distribution

PDF ('GEOMETRIC', m, p)

Arguments

m

is a numeric constant, variable, or expression that specifies the number of failures before the first success.

Range

m ≥ 0

p

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

Range

0 ≤ p ≤ 1

Details

The PDF function for the geometric distribution returns the probability density function of a geometric distribution, with the parameter p. The PDF function is evaluated at the value m.

multiline equation line 1 upper P 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 p left-parenthesis 1 minus p right-parenthesis Superscript m Baseline 2nd Column m greater-than-or-equal-to 0 EndLayout

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

Hypergeometric Distribution

PDF ('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 odds ratio parameter.

Range

o > 0

Details

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

StartLayout 1st Row 1st Column upper P 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 StartBinomialOrMatrix upper R Choose x EndBinomialOrMatrix StartBinomialOrMatrix upper N minus upper R Choose n minus x EndBinomialOrMatrix o Superscript x Baseline Over normal upper Sigma 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 0 2nd Column x greater-than m i n left-parenthesis upper R comma n right-parenthesis EndLayout EndLayout

Laplace Distribution

PDF ('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 PDF function for the Laplace distribution returns the probability density function of the Laplace distribution, with the location parameter θ and the scale parameter λ. The PDF function is evaluated at the value x.

multiline equation line 1 upper P 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 StartFraction 1 Over 2 lamda EndFraction e x p left-parenthesis minus StartFraction StartAbsoluteValue x minus theta EndAbsoluteValue Over lamda EndFraction right-parenthesis

Logistic Distribution

PDF ('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 PDF function for the logistic distribution returns the probability density function of a logistic distribution, with the location parameter θ and the scale parameter λ. The PDF function is evaluated at the value x.

multiline equation line 1 upper P 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 StartStartFraction e x p left-parenthesis minus StartFraction x minus theta Over lamda EndFraction right-parenthesis OverOver lamda left-parenthesis 1 plus e x p left-parenthesis minus StartFraction x minus theta Over lamda EndFraction right-parenthesis right-parenthesis squared EndEndFraction

Lognormal Distribution

PDF ('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. exp(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 PDF function for the lognormal distribution returns the probability density function of a lognormal distribution, with the log scale parameter θ and the shape parameter λ. The PDF function is evaluated at the value x.

multiline equation line 1 upper P 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 x lamda RootIndex StartRoot 2 pi EndRoot EndFraction e x p left-parenthesis minus StartFraction left-parenthesis l o g left-parenthesis x right-parenthesis minus theta right-parenthesis squared Over 2 lamda squared EndFraction right-parenthesis 2nd Column x greater-than 0 EndLayout

Negative Binomial Distribution

PDF ('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 PDF function for the negative binomial distribution returns the probability density function of a negative binomial distribution, with probability of success p and number of successes n. The PDF function is evaluated at the value m.

multiline equation line 1 upper P 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 StartBinomialOrMatrix n plus m minus 1 Choose n minus 1 EndBinomialOrMatrix p Superscript n Baseline left-parenthesis 1 minus p right-parenthesis Superscript m 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

PDF ('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 PDF function for the normal distribution returns the probability density function of a normal distribution, with the location parameter θ and the scale parameter λ. The PDF function is evaluated at the value x.

multiline equation line 1 upper P 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 e x p left-parenthesis minus StartFraction left-parenthesis x minus theta right-parenthesis squared Over 2 lamda squared EndFraction right-parenthesis

Normal Mixture Distribution

PDF ('NORMALMIX', x, n, p₁, p₂, ... p_n, m₁, m₂, ... m_n, s₁, s₂, ... s_n)

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 integer number of mixtures.

Range

n=1, 2, ...

p_i

is a list of numeric constants, variables, or expressions that specifies the n proportions, , where .

Range

p=0, 1, ...

m_i

is a list of numeric constants, variables, or expressions that specifies the n means .

s_i

is a list of numeric constants, variables, or expressions that specifies the n standard deviations .

Range

s > 0

Details

The PDF 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 P 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 Superscript prime Baseline 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 P 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 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

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

Arguments

x

is a numeric constant, variable, or expression that specifies a numeric 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 PDF function for the Pareto distribution returns the probability density function of a Pareto distribution, with the shape parameter a and the scale parameter k. The PDF function is evaluated at the value x.

multiline equation line 1 upper P 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 StartFraction a Over k EndFraction left-parenthesis StartFraction k Over x EndFraction right-parenthesis Superscript a plus 1 Baseline 2nd Column x greater-than-or-equal-to k EndLayout

Poisson Distribution

PDF ('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 PDF function for the Poisson distribution returns the probability density function of a Poisson distribution, with mean m. The PDF function is evaluated at the value n.

multiline equation line 1 upper P upper D upper F left-parenthesis prime upper P upper O upper I upper S upper S upper O upper N Superscript prime Baseline 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 e Superscript negative m Baseline StartFraction m Superscript n Baseline Over n 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

PDF ('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 PDF function for the T distribution returns the probability density function of a T distribution, with degrees of freedom df and the noncentrality parameter nc. The PDF function is evaluated at the value x. This PDF function accepts noninteger degrees of freedom. If nc is omitted or equal to zero, the value returned is from the central T distribution. In this equation, let

and let

multiline equation line 1 upper P 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 StartFraction nu Over 2 EndFraction minus 1 Baseline upper Gamma left-parenthesis StartFraction nu Over 2 EndFraction right-parenthesis EndEndFraction integral Subscript 0 Superscript infinity Baseline x Superscript v minus 1 Baseline e Superscript minus one-half x squared Baseline StartFraction 1 Over RootIndex StartRoot 2 pi EndRoot EndFraction e Superscript minus one-half left-parenthesis StartFraction t x Over RootIndex StartRoot v EndRoot EndFraction minus delta right-parenthesis squared Baseline StartFraction x Over RootIndex StartRoot v EndRoot EndFraction d x

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

Tweedie Distribution

PDF ('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 y>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 PDF function for the Tweedie distribution returns an exponential dispersion model with variance and mean related by the equation variance

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 alpha Baseline Over phi Superscript j left-parenthesis 1 minus alpha right-parenthesis Baseline left-parenthesis 2 minus p right-parenthesis Superscript j Baseline j factorial normal upper Gamma left-parenthesis minus j alpha right-parenthesis EndFraction right-parenthesis exp left-parenthesis StartFraction 1 Over phi EndFraction left-parenthesis y StartFraction mu Superscript 1 minus p Baseline minus 1 Over 1 minus p EndFraction minus StartFraction mu Superscript 2 minus p Baseline minus 1 Over 2 minus p EndFraction right-parenthesis right-parenthesis

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

is near 0. In either of these cases, the accuracy might be as low as six digits.

Note: To avoid issues with numerical data, µ and Φ cannot be less than the constant SQRTMACEPS.

Uniform Distribution

PDF ('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 PDF function for the uniform distribution returns the probability density function of a uniform distribution, with the left location parameter l and the right location parameter r. The PDF function is evaluated at the value x.

multiline equation line 1 upper P 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 1 Over r minus l EndFraction 2nd Column l less-than-or-equal-to x less-than-or-equal-to r 3rd Row 1st Column 0 2nd Column x greater-than r EndLayout

Wald (Inverse Gaussian) Distribution

PDF ('WALD', x, λ <, µ>)

PDF ('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 PDF function for the Wald distribution returns the probability density function of a Wald distribution, with the shape parameter λ, which is evaluated at the value x.

f x left-parenthesis x right-parenthesis equals left-bracket StartFraction lamda Over 2 pi x cubed EndFraction right-bracket Superscript 1 slash 2 Baseline exp left-brace minus StartFraction lamda Over 2 mu squared x EndFraction left-parenthesis x minus mu right-parenthesis squared right-brace comma x greater-than 0

Weibull Distribution

PDF('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 PDF function for the Weibull distribution returns the probability density function of a Weibull distribution, with the shape parameter a and the scale parameter λ. The PDF function is evaluated at the value x.

multiline equation line 1 upper P 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 e x p left-parenthesis minus left-parenthesis StartFraction x Over lamda EndFraction right-parenthesis Superscript a Baseline right-parenthesis StartFraction a Over lamda EndFraction left-parenthesis StartFraction x Over lamda EndFraction right-parenthesis Superscript a minus 1 Baseline 2nd Column x greater-than-or-equal-to 0 EndLayout

Example

The following SAS statements produce these results:

SAS Statement	Result
y=pdf('BERN', 0, .25);	0.75
y=pdf('BERN', 1, .25);	0.25
y=pdf('BETA', 0.2, 3, 4);	1.2288
y=pdf('BINOM', 4, .5, 10);	0.20508
y=pdf('CAUCHY', 2);	0.063662
y=pdf('CHISQ', 11.264, 11);	0.081686
y=pdf('CONMAXPOI', .2, 2.3, .4);	0.0097732635
y=pdf('EXPO', 1);	0.36788
y=pdf('F', 3.32, 2, 3);	0.054027
y=pdf('GAMMA', 1, 3);	0.18394
y=pdf('GENPOISSON', 9, 1, .7);	0.0150130915
y=pdf('GEOMETRIC', 5, .3);	y=0.050421
y=pdf('HYPER', 2, 200, 50, 10);	0.28685
y=pdf('LAPLACE', 1);	0.18394
y=pdf('LOGISTIC', 1);	0.19661
y=pdf('LOGNORMAL', 1);	0.39894
y=pdf('NEGB', 1, .5, 2);	0.25
y=pdf('NORMAL', 1.96);	0.058441
y=pdf('NORMALMIX',2.3, 3, .33, .33, .34, .5, 1.5, 2.5, .79, 1.6, 4.3);	0.1166
y=pdf('PARETO', 1, 1);	1
y=pdf('POISSON', 2, 1);	0.18394
y=pdf('T', .9, 5);	0.24194
y=pdf('TWEEDIE', .8, 5);	0.7422908236
y=pdf('UNIFORM', 0.25);	1
y=pdf('WALD', 1, 2);	0.56419
y=pdf('WEIBULL', 1, 2);	0.73576

References

Shmueli, G., T. Minka,, S. Borle, and P. Boatwright. “A Useful Distribution for Fitting Discrete Data: Revival of the Conway-Maxwell-Poisson Distribution.” 2005. Applied Statistics : 54:127–142.

Last updated: March 16, 2017