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.

parameter-1, …, parameter-k

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

See

Details

Generating Random Numbers

Reproducing a Random Number Stream

Duplicate Values in the Mersenne-Twister RNG Algorithm

Bernoulli Distribution

Beta Distribution

Binomial Distribution

Cauchy Distribution

Chi-Square Distribution

Erlang Distribution

Exponential Distribution

F Distribution

Gamma Distribution

Geometric Distribution

Hypergeometric Distribution

Lognormal Distribution

Negative Binomial Distribution

Triangular Distribution

Uniform Distribution

Weibull Distribution

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.

If the initial seed is exactly divisible by 8192, the RAND function uses the 2002 initialization algorithm (Matsumoto and Nishimura 2002). Otherwise, RAND uses the 1998 initialization algorithm.

Reproducing a Random Number Stream

If you want to create reproducible streams of random numbers, 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 nonpositive seed value in the CALL STREAMINIT routine), then RAND obtains an initial seed from the Intel RdRand instruction on machines that support the instruction, or from the current date-time value on machines that do not support RdRand. For more information, see CALL STREAMINIT.Creating a Reproducible Stream of Random Numbers

Duplicate Values in the Mersenne-Twister RNG Algorithm

The Mersenne-Twister RNG algorithm has an extremely long period, but this does not imply that large random samples are devoid of duplicate values. The RAND function returns at most 2³² distinct values. In a random uniform sample of size 10⁵, the chance of drawing at least one duplicate is greater than 50%. The expected number of duplicates in a random uniform sample of size M is approximately M²/2³³ when M is much less than 2³². For example, you should expect about 115 duplicates in a random uniform sample of size M=10⁶. These results are consequences of the famous “birthday matching problem” in probability theory.

Bernoulli Distribution

x=RAND('BERNOULLI', p)

Arguments

x

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

multiline equation line 1 f left-parenthesis x right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column 1 2nd Column p equals 0 comma x equals 0 2nd Row 1st Column p Superscript x Baseline left-parenthesis 1 minus p right-parenthesis Superscript 1 minus x Baseline 2nd Column 0 less-than p less-than 1 comma x equals 0 comma 1 3rd Row 1st Column 1 2nd Column p equals 1 comma x equals 1 EndLayout

Range

x = 0, 1

p

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

Range

0 ≤ p ≤ 1

Beta Distribution

x=RAND('BETA', a, b)

Arguments

x

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

multiline equation line 1 f left-parenthesis x right-parenthesis equals StartFraction upper Gamma left-parenthesis a plus b right-parenthesis Over upper Gamma left-parenthesis a right-parenthesis upper Gamma left-parenthesis b right-parenthesis EndFraction x Superscript a minus 1 Baseline left-parenthesis 1 minus x right-parenthesis Superscript b minus 1

Range

0 < x < 1

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

Binomial Distribution

x=RAND('BINOMIAL', p, n)

Arguments

x

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

Range

x = 0, 1, ..., n

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 = 1, 2, ...

Cauchy Distribution

x=RAND('CAUCHY')

Arguments

x

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

multiline equation line 1 f left-parenthesis x right-parenthesis equals StartFraction 1 Over pi left-parenthesis 1 plus x squared right-parenthesis EndFraction

Range

–∞ < x < ∞

Chi-Square Distribution

x=RAND('CHISQUARE', df)

Arguments

x

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

multiline equation line 1 f left-parenthesis x right-parenthesis equals StartStartFraction 2 Superscript minus d f slash 2 Baseline OverOver upper Gamma left-parenthesis StartFraction d f Over 2 EndFraction right-parenthesis EndEndFraction x Superscript d f slash 2 minus 1 Baseline e Superscript negative x slash 2

Range

x > 0

df

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

Range

df > 0

Erlang Distribution

x=RAND('ERLANG', a)

Arguments

x

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

multiline equation line 1 f left-parenthesis x right-parenthesis equals StartFraction 1 Over upper Gamma left-parenthesis a right-parenthesis EndFraction x Superscript a minus 1 Baseline e Superscript negative x

Range

x > 0

a

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

Range

a = 1, 2, ...

Exponential Distribution

x=RAND('EXPONENTIAL', <σ>)

Arguments

x

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

multiline equation line 1 f left-parenthesis x right-parenthesis equals StartFraction e Superscript negative x slash sigma Baseline Over sigma EndFraction Superscript

Range

x > 0

σ

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

Default

The optional shape parameter σ > 0 has the default value σ = 1.

F Distribution

x=RAND('F', n, d)

Arguments

x

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

multiline equation line 1 f left-parenthesis x right-parenthesis equals StartStartFraction upper Gamma left-parenthesis StartFraction n plus d Over 2 EndFraction right-parenthesis OverOver upper Gamma left-parenthesis StartFraction n Over 2 EndFraction right-parenthesis upper Gamma left-parenthesis StartFraction d Over 2 EndFraction right-parenthesis EndEndFraction StartFraction n Superscript n slash 2 Baseline d Superscript d slash 2 Baseline x Superscript n slash 2 minus 1 Baseline Over left-parenthesis d plus n x right-parenthesis Superscript left-parenthesis n plus d right-parenthesis slash 2 Baseline EndFraction

Range

x > 0

n

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

Range

n > 0

d

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

Range

d > 0

Gamma Distribution

x=RAND('GAMMA', a <,λ>)

Arguments

x

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

multiline equation line 1 f left-parenthesis x right-parenthesis equals StartFraction x Superscript a minus 1 Baseline Over times lamda Superscript a Baseline upper Gamma left-parenthesis a right-parenthesis EndFraction e Superscript negative x slash lamda

Range

x > 0

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 shape parameter.

Default

The optional shape parameter λ > 0 has the default value λ = 1.

Geometric Distribution

x=RAND('GEOMETRIC', p)

Arguments

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:

multiline equation line 1 f left-parenthesis x right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column left-parenthesis 1 minus p right-parenthesis Superscript x minus 1 Baseline p 2nd Column 0 less-than p less-than 1 comma x equals 1 comma 2 comma period period period 2nd Row 1st Column 1 2nd Column p equals 1 comma x equals 1 EndLayout

Range

x = 1, 2, …

p

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

Range

0 < p ≤ 1

Hypergeometric Distribution

x=RAND('HYPER', N, R, n)

Arguments

x

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

multiline equation line 1 f left-parenthesis x right-parenthesis equals StartFraction StartBinomialOrMatrix upper R Choose x EndBinomialOrMatrix StartBinomialOrMatrix upper N minus upper R Choose n minus x EndBinomialOrMatrix Over StartBinomialOrMatrix upper N Choose n EndBinomialOrMatrix EndFraction

Range

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

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

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

x=RAND('LOGNORMAL', <θ, λ>)

Arguments

x

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

multiline equation line 1 f left-parenthesis x right-parenthesis equals StartFraction 1 Over x lamda RootIndex StartRoot 2 pi EndRoot EndFraction times exp left-parenthesis minus StartFraction left-parenthesis l n left-parenthesis x right-parenthesis minus theta right-parenthesis ⁡ squared Over 2 lamda ⁡ squared EndFraction

Range

x > 0

θ

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

Default

The optional log-scale parameter θ has the default value θ = 0.

λ

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

Default

The optional log-scale parameter λ has the default value λ = 1.

Negative Binomial Distribution

x=RAND('NEGBINOMIAL', p, k)

Arguments

x

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

multiline equation line 1 f left-parenthesis x right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column StartBinomialOrMatrix x plus k minus 1 Choose k minus 1 EndBinomialOrMatrix left-parenthesis 1 minus p right-parenthesis Superscript x Baseline p Superscript k Baseline 2nd Column 0 less-than p less-than 1 comma x equals 0 comma 1 comma period period period 2nd Row 1st Column 1 2nd Column p equals 1 comma x equals 0 EndLayout

Range

x = 0, 1, ...

k

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

Range

k = 1, 2, ...

p

is a numeric constant, variable, or expression that specifies the 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',<θ, λ>)

Arguments

x

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

multiline equation line 1 f left-parenthesis x 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

Range

–∞ < x < ∞

θ

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

Default

λ

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

Default	1
Range	λ > 0

Poisson Distribution

x=RAND('POISSON', m)

Arguments

x

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

multiline equation line 1 f left-parenthesis x right-parenthesis equals StartFraction m Superscript x Baseline e Superscript negative m Baseline Over x factorial EndFraction

Range

x = 0, 1, ...

m

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

Range

m > 0

T Distribution

x=RAND('T', df)

Arguments

x

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

multiline equation line 1 f left-parenthesis x right-parenthesis equals StartStartFraction upper Gamma left-parenthesis StartFraction d f plus 1 Over 2 EndFraction right-parenthesis OverOver RootIndex StartRoot d f pi EndRoot upper Gamma left-parenthesis StartFraction d f Over 2 EndFraction right-parenthesis EndEndFraction left-parenthesis 1 plus StartFraction x squared Over d f EndFraction right-parenthesis Superscript minus StartFraction d f plus 1 Over 2 EndFraction

Range

–∞ < x < ∞

df

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

Range

df > 0

Tabled Distribution

x=RAND('TABLE', p1, p2, ...)

Arguments

x

is an integer observation from one of the following distributions:

normal upper Sigma Underscript i equals 1 Overscript n Endscripts p Subscript i Baseline less-than 1

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

multiline equation line 1 f left-parenthesis i right-parenthesis equals p Subscript i Baseline comma i equals 1 comma 2 comma ellipsis comma n line 2 a n d line 3 f left-parenthesis n plus 1 right-parenthesis equals 1 minus normal upper Sigma Underscript i equals 1 Overscript n Endscripts p Subscript i

normal upper Sigma Underscript i equals 1 Overscript n Endscripts p Subscript i Baseline greater-than-or-equal-to 1

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

multiline equation line 1 f left-parenthesis i right-parenthesis equals p Subscript i Baseline comma i equals 1 comma 2 comma ellipsis comma n minus 1 line 2 a n d line 3 f left-parenthesis n right-parenthesis equals 1 minus normal upper Sigma Underscript i equals 1 Overscript n minus 1 Endscripts p Subscript i

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

x=RAND('TRIANGLE', h)

Arguments

x

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

StartLayout 1st Row 1st Column f left-parenthesis x right-parenthesis equals 2nd Column StartLayout Enlarged left-brace 1st Row 1st Column StartFraction 2 x Over h EndFraction 2nd Column 0 less-than-or-equal-to x less-than-or-equal-to h 2nd Row 1st Column StartFraction 2 left-parenthesis 1 minus x right-parenthesis Over 1 minus h EndFraction 2nd Column h less-than x less-than-or-equal-to 1 EndLayout EndLayout

In this equation, 0 ≤ h ≤ 1.

Range	0 ≤ x ≤ 1
Note	The distribution can be easily shifted and scaled.

h

is a numeric constant, variable, or expression that specifies the horizontal location of the peak of the triangle.

Range

0 ≤ h ≤ 1

Uniform Distribution

x=RAND('UNIFORM', <a, b>)

Arguments

x

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

Range

The range of x is a ≤ x ≤b. The parameters a and b default values are a=0 and b=1. You must specify values for both a and b if you do not want to use the default values.

a

specifies a value in the probability density function.

Default

b

specifies a value in the probability density function.

Default

The uniform random number generator that the RAND function uses is the Mersenne-Twister (Matsumoto and Nishimura 1998). This generator has a period of

2 Superscript 1 9 9 3 7 Baseline minus 1

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

x=RAND('WEIBULL', a, b)

Arguments

x

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

multiline equation line 1 f left-parenthesis x right-parenthesis equals StartFraction a Over b Superscript a Baseline EndFraction x Superscript a minus 1 Baseline e Superscript minus left-parenthesis StartFraction x Over b EndFraction right-parenthesis Super Superscript a

Range

x ≥ 0

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 scale parameter.

Range

b > 0

Example

The following SAS statements produce these results:

SAS Statement	Result
x=rand('BERN', .75);	0
x=rand('BETA', 3, 0.1);	.99920
x=rand('BINOM', 0.75, 10);	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

References

Matsumoto, M., and T. Nishimura. “Mersenne Twister with Improved Initialization.” 2002. Available Mersenne Twister Home Page.

Fishman, G. S. 1996. Monte Carlo: Concepts, Algorithms, and Applications. New York, USA: Springer-Verlag.

Fushimi, M., and S. Tezuka. “The k-Distribution of Generalized Feedback Shift Register Pseudorandom Numbers.” 1983. Communications of the ACM 26: 516–523.

Gentle, J. E. 1998. Random Number Generation and Monte Carlo Methods. New York, USA: Springer-Verlag.

Lewis, T. G., and W. H. Payne. “Generalized Feedback Shift Register Pseudorandom Number Algorithm.” 1973. Journal of the ACM 20: 456-468.

Matsumoto, M., and Y. Kurita. “Twisted GFSR Generators.” 1992. ACM Transactions on Modeling and Computer Simulation 2: 179–194.

Matsumoto, M., and Y. Kurita. “Twisted GFSR Generators II.” 1994. ACM Transactions on Modeling and Computer Simulation 4: 254–266.

Matsumoto, M., and T. Nishimura. “Mersenne Twister: A 623–Dimensionally Equidistributed Uniform Pseudo-Random Number Generator.” 1998. ACM Transactions on Modeling and Computer Simulation 8: 3–30.

Ripley, B. D. 1987. Stochastic Simulation. New York, USA: Wiley.

Robert, C. P., and G. Casella. 1999. Monte Carlo Statistical Methods. New York, USA: Springer-Verlag.

Ross, S. M. 1997. Simulation. San Diego, USA: Academic Press.

Last updated: March 16, 2017