SAS 9.1 Language Reference Dictionary, Volumes 1, 2 and 3

Computes cumulative distribution functions

Category: Probability

Syntax

CDF ( 'dist',quantile, < ,parm-1, ... ,parm-k >)

Arguments

'dist'

• is a character string that identifies the distribution. Valid distributions are as follows :

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

quantile

• is a numeric random variable.

parm-1, ... ,parm-k

• are optional shape , location , or scale parameters appropriate for the specific distribution.

• See: 'Details' on page 419 for complete information about these parameters

Details

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

Bernoulli Distribution

• CDF ('BERNOULLI', x , p )

where

• x

  • is a numeric random variable.

• p

  • is a numeric probability of success.

  • Range: ‰ p ‰ 1

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 . The equation follows:

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

Beta Distribution

• CDF ('BETA', x , a , b <, l , r >)

where

• x

  • is a numeric random variable.

• a

  • is a numeric shape parameter.

  • Range: a > 0

• b

  • is a numeric shape parameter.

  • Range: b > 0

• l

  • is the numeric left location parameter.

  • Default:

• r

  • is the right location parameter.

  • Default: 1

  • Range: r > l

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 x . The following equation describes the CDF function of the beta distribution:

where

and

Binomial Distribution

• CDF ('BINOMIAL', m , p , n )

where

• m

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

  • Range: m = 0, 1, ...

• p

  • is a numeric probability of success.

  • Range: ‰ p ‰ 1

• n

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

  • Range: n = 0, 1, ...

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 . The equation follows:

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

Cauchy Distribution

• CDF ('CAUCHY', x <, , » >)

where

• x

  • is a numeric random variable.

• • is a numeric location parameter.

  • Default:

• »

  • is a numeric scale parameter.

  • Default: 1

  • Range: » > 0

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

Chi-Square Distribution

• CDF ('CHISQUARE', x , df <, nc >)

where

• x

  • is a numeric random variable.

• df

  • is a numeric degrees of freedom parameter.

  • Range: df > 0

• nc

  • is an optional numeric non- centrality parameter.

  • Range: nc ‰

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

where P _c (.,.) denotes the probability from the central chi-square distribution:

and where P _g ( y , b ) is the probability from the Gamma distribution given by

Exponential Distribution

• CDF ('EXPONENTIAL', x <, » >)

where

• x

  • is a numeric random variable.

• »

  • is a scale parameter.

  • Default: 1

  • Range: » > 0

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

F Distribution

• CDF ('F', x , ndf , ddf <, nc >)

where

• x

  • is a numeric random variable.

• ndf

  • is a numeric numerator degrees of freedom parameter.

  • Range: ndf > 0

• ddf

  • is a numeric denominator degrees of freedom parameter.

  • Range: ddf > 0

• nc

  • is a numeric non-centrality parameter.

  • Range: nc ‰

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 non-centrality parameter nc , is less than or equal to x . This function accepts non-integer degrees of freedom for ndf and ddf . If nc is omitted or equal to zero, the value returned is from a central F distribution. The following equation describes the CDF function of the F distribution:

where P _f ( f , u ₁ , u ₂ ) is the probability from the central F distribution with

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 <, » >)

where

• x

  • is a numeric random variable.

• a

  • is a numeric shape parameter.

  • Range: a > 0

• »

  • is a numeric scale parameter.

  • Default: 1

  • Range: » > 0

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

Geometric Distribution

• CDF ('GEOMETRIC', m,p )

where

• m

  • is a numeric random variable that denotes the number of failures.

  • Range: m = 0, 1, ...

• p

  • is a numeric probability of success.

  • Range: ‰ p ‰ 1

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

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

Hypergeometric Distribution

• CDF ('HYPER', x,N,R,n <, o >)

where

• x

  • is an integer random variable.

• N

  • is an integer population size parameter.

  • Range: N = 1, 2, ...

• R

  • is an integer number of items in the category of interest.

  • Range: R = 0, 1, ..., N

• n

  • is an integer sample size parameter.

  • Range: n = 1, 2, ..., N

• o

  • is an optional numeric odds ratio parameter.

  • Range: o > 0

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. The equation follows:

Laplace Distribution

• CDF ('LAPLACE', x <, , >)

where

• x

  • is a numeric random variable.

• • is a numeric location parameter.

  • Default:

• »

  • is a numeric scale parameter.

  • Default: 1

  • Range: » > 0

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

Logistic Distribution

• CDF ('LOGISTIC', x <, , » >)

where

• x

  • is a numeric random variable.

• • is a numeric location parameter

  • Default:

• »

  • is a numeric scale parameter.

  • Default: 1

  • Range: » > 0

The CDF function for the logistic distribution returns the probability that an observation from a logistic distribution, with a location parameter and a scale parameter » , is less than or equal to x . The equation follows:

Lognormal Distribution

• CDF ('LOGNORMAL', x <, , » >)

where

• x

  • is a numeric random variable.

• • is a numeric location parameter.

  • Default:

• »

  • is a numeric scale parameter.

  • Default: 1

  • Range: » > 0

The CDF function for the lognormal distribution returns the probability that an observation from a lognormal distribution, with the location parameter and the scale parameter » , is less than or equal to x . The equation follows:

Negative Binomial Distribution

• CDF ('NEGBINOMIAL', m,p,n )

where

• m

  • is a positive integer random variable that counts the number of failures.

  • Range: m = 0, 1, ...

• p

  • is a numeric probability of success.

  • Range: ‰ p ‰ 1

• n

  • is an integer parameter that counts the number of successes.

  • Range: n = 1, 2, ...

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

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

Normal Distribution

• CDF ('NORMAL', x <, , » >)

where

• x

  • is a numeric random variable.

• • is a numeric location parameter.

  • Default:

• »

  • is a numeric scale parameter.

  • Default: 1

  • Range: » > 0

The CDF function for the normal distribution returns the probability that an observation from the normal distribution, with the location parameter and the scale parameter » , is less than or equal to x . The equation follows:

Normal Mixture Distribution

• CDF ('NORMALMIX', x,n,p,m,s )

where

• x

  • is a numeric random variable.

• n

  • is the integer number of mixtures.

  • Range: n = 1, 2, ...

• p

  • is the n proportions , p ₁ , p ₂ , p _n , where .

  • Range: p = 0, 1, ...

• m

  • is the n means m ₁ , m ₂ ,...., m _n .

• s

  • is the n standard deviations s ₁ , s ₂ ,...., s _n .

  • Range: s > 0

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 . The equation follows:

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

Pareto Distribution

• CDF ('PARETO', x,a <, k >)

where

• x

  • is a numeric random variable.

• a

  • is a numeric shape parameter.

  • Range: a > 0

• k

  • is a numeric scale parameter.

  • Default: 1

  • Range: k > 0

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 . The equation follows:

Poisson Distribution

• CDF ('POISSON', n,m )

where

• n

  • is an integer random variable.

  • Range: n = 0,1,...

• m

  • is a numeric mean parameter.

  • Range: m > 0

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 . The equation follows:

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

T Distribution

• CDF ('T', t,df <, nc >)

where

• t

  • is a numeric random variable.

• df

  • is a numeric degrees of freedom parameter.

  • Range: df > 0

• nc

  • is an optional numeric non-centrality parameter.

The CDF function for the T distribution returns the probability that an observation from a T distribution, with degrees of freedom df and non-centrality 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 T distribution. The equation follows:

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

Uniform Distribution

• CDF ('UNIFORM', x <, l,r >)

where

• x

  • is a numeric random variable.

• l

  • is the numeric left location parameter.

  • Default:

• r

  • is the numeric right location parameter.

  • Default: 1

  • Range: r > l

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 . The equation follows:

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

Wald (Inverse Gaussian) Distribution

• CDF ('WALD', x,d )

• CDF ('IGAUSS', x,d )

where

• x

  • is a numeric random variable.

• d

  • is a numeric shape parameter.

  • Range: d > 0

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

where (.) denotes the probability from the standard normal distribution.

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

Weibull Distribution

• CDF ('WEIBULL', x, a <, » >)

where

• x

  • is a numeric random variable.

• a

  • is a numeric shape parameter.

  • Range: a > 0

• »

  • is a numeric scale parameter.

  • Default: 1

  • Range: » > 0

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 » is less than or equal to x . The equation follows:

Examples