Page 184 - Fundamentals of Probability and Statistics for Engineers

Some Important Discrete Distributions 167

Answer: for k min (n 1 , m), we have
P X ˆ k \ X ‡ Y ˆ m†
P X ˆ kjX ‡ Y ˆ m†ˆ
P X ‡ Y ˆ m†
P X ˆ k \ Y ˆ m k† P X ˆ k†P Y ˆ m k†
ˆ ˆ
P X ‡ Y ˆ m† P X ‡ Y ˆ m†

n 1 k n 1 k n 2 m k n 2 m‡k
p 1 p† p 1 p†
ˆ k m k
n 1 ‡ n 2 m n 1 ‡n 2 m
p 1 p†
m

n 1 n 2 n 1 ‡ n 2
ˆ ; k ˆ 0; 1; ... ; min n 1 ; m†; 6:12†
k m k m
where we have used the result given in Example 6.3 that X Y is binomially
‡
distributed with parameters (n 1 ‡ n 2 , p).
The distribution given by Equation (6.12) is known as the hypergeometric
distribution. It arises as distributions in such cases as the number of black balls
that are chosen when a sample of m balls is randomly selected from a lot of
n items having n 1 black balls and n 2 white balls (n 1 ‡ n 2 ˆ n ). Let random
variable Z be this number. We have, from Equation (6.12), on replacing n 2
by n n 1 ,

n 1 n n 1 n
p k†ˆ ; k ˆ 0; 1; ... ; min n 1 ; m†: 6:13†
Z
k m k m
6.1.2 GEOMETRIC DISTRIBUTION

Another event of interest arising from Bernoulli trials is the number of trials to
(and including) the first occurrence of success. If X is used to represent this
number, it is a discrete random variable with possible integer values ranging
from one to infinity. Its pmf is easily computed to be

p k†ˆ P FF ... F S†ˆ P F†P F† .. . P F† P S†
X
|‚‚‚‚‚{z‚‚‚‚‚}
|‚‚‚‚‚‚‚‚‚‚‚‚‚‚{z‚‚‚‚‚‚‚‚‚‚‚‚‚‚}
k 1 k 1 6:14†
ˆ q k 1 p; k ˆ 1; 2; ... :
This distribution is known as the geometric distribution with parameter p,
where the name stems from its similarity to the familiar terms in geometric
progression. A plot of p (k) is given in Figure 6.1.
X

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