Download SAS Functions Useful for Probability Computations

SAS Functions Useful For Probability Computations
N. D. Prabhakar
General Motors Corporation
The first argument for each of these
INTRODUCTION:
functions is a numeric code which specifies
the particular
This paper describes a group of eight SAS
functions that are being developed, which
can evaluate, the following eignt functions
or qUBntities associated with a univariate
probability distribution :
CDF
PDF
CHF
MGF
CDFINV
RAWMMT
CNTIJIIMT
RNll>1
~
of probability
distributions. For example, a code of 1
means a binomial distribution, a code of 50
means a normal distribution and a code of
102 means a Poisson sum of binomial
distributions. A partial list of codes and
types is shown in table 1
the oumulative distribution
function,
the probability densit,y
function,
the characteristic function,
the moment ~nerating function,
the inverse of the cumulative
distribution function
The next group of arguments for these
functions specif.y the parameters of the
particular type of distribution and thereey
a unique probability distribution on the
real line. The actual number of parameters
and their meaning depend on the type
specified by the first ar~ment of the
functions. For instance, the parameters for
the binomial distribution are the number of
trials (n) and the chance for success in a
trial (p), whereas the parameters for the
normal distribution are the mean and
variance. The parameters are seperated by
commas and must appear in their correct
order. Table 1 shows the parameters for the
different ~pes and their meaning.
(percentage points).
the raw moments,
the central moments,
a random observation from the
distribution.
The probability distributions for which the
above functions can be evaluated will
include a wide class of distributions such
as (1) basic discrete and continuous
distributions - e.g. binomial, poisson,
hypergeometric, negative binomial,
logarithmic series, normal, lognormal,
cauchy, uniform, exponential, €f3l1lID3., chisquare, Student's t, Beta, Fisher's F,
Weibullj (2) random sum distributions of
some basic discrete distributions - e.g.
poisson sum binomial, binomdal sum poisson
etc; (3) certain compound distributions
where the base distribution is discrete e.g. Neyman's type A; (4) certain
noncentral continuous distributions - e.g.
noncentral t, F, chi-square.
The last.argwrnent for each function is a
number a. Given the ~pe and parameters,
one has a specific probabili~ distribution
on the real line. Let X be a random
variable with this distrib~tion. Then the
value returned Qy CDF is the probability
P(X<=a). The value returned
by
PDF is f(a)
where f is the probabili~ (densi~)
runction of X. The value returned b,y CHF is
the complex number E(exp( -1 aX where E
stands for the expected value. The value
renlrnerl
by
MGF is
E(exp(~
X)). The value
returned by CDFINV is the smallest value b
such that P(X<=b) = a. The value returned
by RAWMMT is E(X**a) and that returned ey
CNTlMMT is E( (X-E(X))**a). The function
FUNCTIONS:
RNDM returns a random observation from the
distribution of X, using a seed of a, where
appropriate.
The eignt SAS :unctions called CDF, PDF,
CHF, MGF, CDFllW, RAWMMT, CNTIMMT and RNIM
will compute the cumulative distribution
function, the probability density function,
the characteristic fUnction, the moment
generating function, the inverse of the
cumulative distribution function, the raw
moment, the central moments and a random
observation respectively for the
distribution specified among the arguments
for the functions.
The performance characteristics of the
algorithms used for the function
evaluations are currently being studied.
CONCLUSION:
ComputatiOns involving distribution
functiOns, percentage pOints etc. of
various univariate distributions arise ver,y
frequent~ in areas of applied probability
suet as queueing theory, reliability thco~
etc. Also random observatiOns from
different univariate distrib~tions are
required for use in simulation studies. In
such contexts, the above SAS fUnctions can
be useful in aLl@Il8nting the currently
existing probability and random number
generating functions of SAS.
016
REFffiENCE:
SAS User's Guide: Basics (1982)
SAS Institute Inc., Cary, NC.
Collected Algorithms,
ACM Vol 1 ,2,3
Tables of the F & related distributions
with algorithms.
K. V. Mardia &
P. J. Zemroch
Academic Press, 1978
Formulae & tables for Statistical work.
Ed. by HaD, Mitra, MathRi, Ramamurt~'1Y
Statistical Publishing Society,
1975
For fUrther ionformation please contact:
General Motors Corporation
Dr. N.D. Prabhakar
485 W. Milwaukee
A-104D
Detroit, Mi 48202
(313) 556-3148
TYPE
CODE
Binomial
Hypergeometric
Poisson
Negative Binomial
Logarithmic Series
1
2
3
4
n,p
N,M,n
m
n,p
p
m, '
b+b
5
50
51 m, ,
52 a,b
53
.,b
54
55 n,_
56 m
57 m
58 m,n
59 m,n
100 N,c,k,p
pH
102 m,k.p
Normal
Lognorma.l
Cauchy
Uniform
Exponential
Gamma
Chisquare
Student's t
Beta
Fisher's F
MEANIYG OR DENSITY FUNCTION
PARAMETERS
•
202 ',0
noncent ral chisq. 300 m,a
pp
Doncentral t
301 m,a
noncentral F
302 m,a,n
number of trials,chance of success
C(M,x)*C(N-M,n-x)/C(N,n)
[m**x]*exp(-m)/m!
C(n-x+l.x)*[p**x]/[(l+p)**(n+x)1
-(p**x]/[x*log(l-p)]
mean,s.d.
log(y) where y is normal m.s
a/[pi*(a*a+«x-b)**2)]
1/ (b-a)
a*exp(-a*x)
[(a**n)(x**(n-l»exp(-a*x)]/gamma(n)
degrees of freedom
degrees of freedom
[(x**(m-l»«l-x)**(n-l»]/beta(m,n)
numerator, denominator d.f.
sum of n binomial k,p where
n is binomial N,~
sum of n binomial k.p where
n is poisson m
poisson m where m/c is poisson a
degrees of freedom,
noncentrality parameter
y/sqrt(z/m) where y is normal a,l
and z is chisquare m
(y/m)/(z/n) where y is noncentral
chisquare m,a ann z is chisquare n
817