RAND Function

Arguments

dist

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
Erlang	ERLANG
Exponential	EXPONENTIAL
F	F
Gamma	GAMMA
Geometric	GEOMETRIC
Hypergeometric	HYPERGEOMETRIC
Lognormal	LOGNORMAL
Negative binomial	NEGBINOMIAL
Normal	NORMAL|GAUSSIAN
Poisson	POISSON
T	T
Tabled	TABLE
Triangular	TRIANGLE
Uniform	UNIFORM
Weibull	WEIBULL

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

parm-1,...,parm-k

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

See:	Details for complete information about these parameters

Generating Random Numbers

The RAND function generates random numbers from various continuous and discrete distributions. Wherever possible, the simplest form of the distribution is used.

The RAND function uses the Mersenne-Twister random number generator (RNG) that was developed by Matsumoto and Nishimura (1998). The random number generator has a very long period (2¹⁹⁹³⁷ - 1) and very good statistical properties. The period is a Mersenne prime, which contributes to the naming of the RNG. The algorithm is a twisted generalized feedback shift register (TGFSR) that explains the latter part of the name. The TGFSR gives the RNG a very high order of equidistribution (623-dimensional with 32-bit accuracy), which means that there is a very small correlation between successive vectors of 623 pseudo-random numbers.

The RAND function is started with a single seed. However, the state of the process cannot be captured by a single seed. You cannot stop and restart the generator from its stopping point.

Reproducing a Random Number Stream

If you want to create reproducible streams of random numbers, then use the CALL STREAMINIT routine to specify a seed value for random number generation. Use the CALL STREAMINIT routine once per DATA step before any invocation of the RAND function. If you omit the call to the CALL STREAMINIT routine (or if you specify a non-positive seed value in the CALL STREAMINIT routine), then RAND uses a call to the system clock to seed itself. For more information, see CALL STREAMINIT Creating a Reproducible Stream of Random Numbers.

Bernoulli Distribution

where

x

is an observation from the distribution with the following probability density function:

Range:

x = 0, 1

p

is a numeric probability of success.

Range:

0 p 1

Beta Distribution

where

x

is an observation from the distribution with the following probability density function:

Range:

0 < x < 1

a

is a numeric shape parameter.

Range:

a > 0

b

is a numeric shape parameter.

Range:

b > 0

Binomial Distribution

where

x

is an integer observation from the distribution with the following probability density function:

Range:

x = 0, 1, ..., n

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 = 1, 2, ...

Cauchy Distribution

where

x

is an observation from the distribution with the following probability density function:

Range:

- < x <

Chi-Square Distribution

where

x

is an observation from the distribution with the following probability density function:

Range:

x > 0

df

is a numeric degrees of freedom parameter.

Range:

df > 0

Erlang Distribution

where

x

is an observation from the distribution with the following probability density function:

Range:

x > 0

a

is an integer numeric shape parameter.

Range:

a = 1, 2, ...

Exponential Distribution

where

x

is an observation from the distribution with the following probability density function:

Range:

x > 0

F Distribution

where

x

is an observation from the distribution with the following probability density function:

Range:

x > 0

ndf

is a numeric numerator degrees of freedom parameter.

Range:

ndf > 0

ddf

is a numeric denominator degrees of freedom parameter.

Range:

ddf > 0

Gamma Distribution

where

x

is an observation from the distribution with the following probability density function:

Range:

x > 0

a

is a numeric shape parameter.

Range:

a > 0

Geometric Distribution

where

x

is an integer count that denotes the number of trials that are needed to obtain one success. X is an integer observation from the distribution with the following probability density function:

Range:

x = 1, 2, ...

p

is a numeric probability of success.

Range:

0 < p 1

Hypergeometric Distribution

where

x

is an integer observation from the distribution with the following probability density function:

Range:

x = max(0, (n - (N - R))), ..., min(n, R)

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

The hypergeometric distribution is a mathematical formalization of an experiment in which you draw n balls from an urn that contains N balls, R of which are red. The hypergeometric distribution is the distribution of the number of red balls in the sample of n.

Lognormal Distribution

where

x

is an observation from the distribution with the following probability density function:

Range:

x > 0

Negative Binomial Distribution

where

x

is an integer observation from the distribution with the following probability density function:

Range:

x = 0, 1, ...

k

is an integer parameter that is the number of successes. However, non-integer k values are allowed as well.

Range:

k = 1, 2, ...

p

is a numeric probability of success.

Range:

0 < p 1

The negative binomial distribution is the distribution of the number of failures before k successes occur in sequential independent trials, all with the same probability of success, p.

Normal Distribution

x = RAND('NORMAL',<,,>)

where

x

is an observation from the normal distribution with a mean of and a standard deviation of , that has the following probability density function:

Range:

- < x <

is the mean parameter.

Default:

0

is the standard deviation parameter.

Default:	1
Range:	> 0

Poisson Distribution

where

x

is an integer observation from the distribution with the following probability density function:

Range:

x = 0, 1, ...

m

is a numeric mean parameter.

Range:

m > 0

T Distribution

where

x

is an observation from the distribution with the following probability density function:

Range:

- < x <

df

is a numeric degrees of freedom parameter.

Range:

df > 0

Tabled Distribution

where

x

is an integer observation from one of the following distributions:

If , then x is an observation from this probability density function:

If for some index , then x is an observation from this probability density function:

p1, p2, ...

are numeric probability values.

Range:	0 p1, p2, ... 1
Restriction:	The maximum number of probability parameters depends on your operating environment, but the maximum number of parameters is at least 32,767.

The tabled distribution takes on the values 1, 2, ..., n with specified probabilities.

Note:   By using the FORMAT statement, you can map the set {1, 2, ..., n} to any set of n or fewer elements.  

Triangular Distribution

where

x

is an observation from the distribution with the following probability density function:

where 0 h 1.

Range:

0 x 1

Note:   The distribution can be easily shifted and scaled.  

h

is the horizontal location of the peak of the triangle.

Range:

0 h 1

Uniform Distribution

where

x

is an observation from the distribution with the following probability density function:

Range:

0 < x < 1

The uniform random number generator that the RAND function uses is the Mersenne-Twister (Matsumoto and Nishimura 1998). This generator has a period of and 623-dimensional equidistribution up to 32-bit accuracy. This algorithm underlies the generators for the other available distributions in the RAND function.

Weibull Distribution

where

x

is an observation from the distribution with the following probability density function:

Range:

x 0

a

is a numeric shape parameter.

Range:

a > 0

b

is a numeric scale parameter.

Range:

b > 0

x=rand('BERN',.75);

0

x=rand('BETA',3,0.1);

.99920

x=rand('BINOM',10,0.75);

10

x=rand('CAUCHY');

-1.41525

x=rand('CHISQ',22);

25.8526

x=rand('ERLANG', 7);

7.67039

x=rand('EXPO');

1.48847

x=rand('F',12,322);

1.99647

x=rand('GAMMA',7.25);

6.59588

x=rand('GEOM',0.02);

43

x=rand('HYPER',10,3,5);

1

x=rand('LOGN');

0.66522

x=rand('NEGB',0.8,5);

33

x=rand('NORMAL');

1.03507

x=rand('POISSON',6.1);

6

x=rand('T',4);

2.44646

x=rand('TABLE',.2,.5);

2

x=rand('TRIANGLE',0.7);

.63811

x=rand('UNIFORM');

.96234

x=rand('WEIB',0.25,2.1);

6.55778

Copyright © 2011 by SAS Institute Inc., Cary, NC, USA. All rights reserved.