Functions and CALL Routines

# CDF Function

Returns a value from a cumulative probability distribution.
 Category: Probability

## Syntax

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

### 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
Exponential EXPONENTIAL
F F
Gamma GAMMA
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
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.

parm-1, ... ,parm-k

are optional constants, variables, or expressions that specify shape, location, or scale parameters 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.

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 .

## Syntax

 CDF('BERNOULLI',x,p)

where

x

is a numeric random variable.

p

is a numeric probability of success.

 Range: 0 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.

## Syntax

 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: 0
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 v. The following equation describes the CDF function of the beta distribution:

where

and

## Syntax

 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: 0 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.

## Syntax

 CDF('CAUCHY',x<,,>)

where

x

is a numeric random variable.

is a numeric location parameter.

 Default: 0

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:

## Syntax

 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 0

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. In the following equation, let $\nu$ = df and let $\lambda$ = nc. The following equation describes the CDF function of the chi-square distribution:

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

and where Pg(y,b) is the probability from the gamma distribution given by

## Syntax

 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:

## Syntax

 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 0

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. In the following equation, let $\nu_1$ = ndf, let $\nu_2$ = ddf, and let $\lambda$ = nc. The following equation describes the CDF function of the F distribution:

where Pf(f,u1,u2) is the probability from the central F distribution with

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

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

## Syntax

 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:

## Syntax

 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: 0 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.

## Syntax

 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:

## Syntax

 CDF('LAPLACE',x<,,>)

where

x

is a numeric random variable.

is a numeric location parameter.

 Default: 0

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:

## Syntax

 CDF('LOGISTIC',x<,,>)

where

x

is a numeric random variable.

is a numeric location parameter

 Default: 0

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:

## Syntax

 CDF('LOGNORMAL',x<,,>)

where

x

is a numeric random variable.

specifies a numeric log scale parameter. (exp() is a scale parameter.)

 Default: 0

specifies a numeric shape parameter.

 Default: 1 Range: > 0

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

## Syntax

 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: 0 p 1
n

is a numeric value that counts the number of successes.

 Range: n > 0

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.

## Syntax

 CDF('NORMAL',x<,,>)

where

x

is a numeric random variable.

is a numeric location parameter.

 Default: 0

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:

## Syntax

 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, , where .

 Range: p = 0, 1, ...
m

is the n means .

s

is the n standard deviations .

 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.

## Syntax

 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:

## Syntax

 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.

## Syntax

 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. In the following equation, let $\nu$ = df and let $\delta$ = nc. The equation follows:

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

## Syntax

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

where

x

is a numeric random variable.

l

is the numeric left location parameter.

 Default: 0
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.

## Syntax

 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.

## Syntax

 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:

SAS Statements Results
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('EXPO',1);
0.63212
y=cdf('F',3.32,2,3);
0.82639
y=cdf('GAMMA',1,3);
0.080301
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('UNIFORM',0.25);
0.25
y=cdf('WALD',1,2);
0.62770
y=cdf('WEIBULL',1,2);
0.63212

 Functions:

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