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184 Fundamentals of Probability and Statistics for Engineers

Table 6.3 Summary of discrete distributions
Distribution Probability mass function Parameters Mean Variance
n

k
Binomial p (1 p) n k , n, p np np(1 p)
k
k ˆ 0, 1, .. . , n
n
. mn 1 (n n 1 )(n m)
n 1 n n 1 mn 1
Hypergeometric , n, n 1 , m
2
k m k m n n (n 1)
k ˆ 0, 1, .. . , min (n 1 , m)
1 1 p
Geometric (1 p) k 1 p, p 2
k ˆ 1, 2, .. . p p

k 1 r r(1 p)
r
Negative binomial p (1 p) k r , r, p 2
(Pascal) r 1 p p
k ˆ r, r ‡ 1, .. .
n!
k r
Multinomial p p p , n, p i , i ˆ 1, , rnp i np i (1 p i )
k 1 k 2
2
1
r
k 1 !k 2 ! k r !
k 1 , .. . , k r ˆ 0, 1, 2, ... ,
P r P r
k i ˆ n, p i ˆ 1,
iˆ1 iˆ1
i ˆ 1, .. . , r
k t
( t) e
Poisson , k ˆ 0, 1, 2, .. . t t t
k!
FURTHER READING

Clark, R.D., 1946, ‘‘An Application of the Poisson Distribution’’, J. Inst. Actuaries 72 48–52.
Solloway, C.B., 1993, ‘‘A Simplified Statistical Model for Missile Launching: III’’, TM
312–287, Jet Propulsion Laboratory, Pasadena, CA.
Binomial and Poisson distributions are widely tabulated in the literature. Additional
references in which these tables can be found are:
Arkin, H., and Colton, R. 1963, Tables for Statisticians, 2nd eds, Barnes and Noble,
New York.
Beyer, W.H., 1966, Handbook of Tables for Probability and Statistics, Chemical Rubber
Co., Cleveland, OH.
Grant, E.C., 1964, Statistical Quality Control, 3rd eds, McGraw-Hill, New York.
Haight, F.A., 1967, Handbook of the Poisson Distribution, John Wiley & Sons Inc.,
New York.
Hald, A., 1952, Statistical Tables and Formulas, John Wiley & Sons Inc., New York.
Molina, E.C., 1949, Poisson’s Exponential Binomial Limit, Von Nostrand, New York.
National Bureau of Standards, 1949, Tables of the Binomial Probability Distributions:
Applied Mathematics Series 6, US Government Printing Office, Washington, DC.
Owen, D., 1962, Handbook of Statistical Tables, Addison-Wesley, Reading, MA.
Pearson, E.S., and Harley, H.O. (eds), 1954, Biometrika Tables for Statisticians, Volume 1,
Cambridge University Press, Cambridge, England.

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