Fundamentals of Probability and Statistics for Engineers

Answer: for k min (n 1 , m), we have

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 su ch 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 ( ). Let random
variable Z be this number. We have, from Equation (6.12), on replacing n 2
by n n 1 ,

6.1.2 G eometric D istribution

AnothereventofinterestarisingfromBernoullitrialsisthenumberoftrialsto
(and including) the first occurrence of success. If X is used to represent this
number,itisadiscreterandomvariablewithpossibleintegervaluesranging
fromonetoinfinity.Itspmfiseasilycomputedtobe

This distribution is known as the geometric distribution with parameter p,
wherethenamestemsfromitssimilaritytothefamiliartermsingeometric
progression. A plot of pX(k) is given in Figure 6 .1.

SomeImportantDiscreteDistributions 167



P…XˆkjX‡Yˆm†ˆ
P…Xˆk\X‡Yˆm†
P…X‡Yˆm†

ˆ

P…Xˆk\Yˆmk†

P…X‡Yˆm†

ˆ

P…Xˆk†P…Yˆmk†

P…X‡Yˆm†

ˆ

n 1

k



pk… 1 p†n^1 k

n 2

mk



pmk… 1 p†n^2 m‡k

n 1 ‡n 2

m



pm… 1 p†n^1 ‡n^2 m

ˆ

n 1

k

n 2

mk

n 1 ‡n 2

m



; kˆ 0 ; 1 ;...;min…n 1 ;m†; … 6 : 12 †

‡

‡

n 1 ‡n 2 ˆn

pZ…k†ˆ

n 1

k

nn

1

mk

 n

m



; kˆ 0 ; 1 ;...;min…n 1 ;m†: … 6 : 13 †

pX…k†ˆP…FF|‚‚‚‚‚{z‚‚‚‚...F‚}

k 1

S†ˆP…F†P…F†...P…F†

|‚‚‚‚‚‚‚‚‚‚‚‚‚‚{z‚‚‚‚‚‚‚‚‚‚‚‚‚‚}

k 1

P…S†

ˆqk^1 p; kˆ 1 ; 2 ;...:

… 6 : 14 †