Fundamentals of Probability and Statistics for Engineers

Some Important Discrete Distributions 167Answer: for k min (n 1 , m), we haveP…X ˆ kjX ‡ Y ˆ m†ˆP…X ˆ k \ X ‡ Y ˆ m†P…X ‡ Y ˆ m†P…X ˆ k \ Y ˆ m k† P…X ˆ k†P…Y ˆ m k†ˆ ˆP…X ‡ Y ˆ m† P…X ‡ Y ˆ m†n1p k …1 p† n1 k n2p m k n2 m‡k…1 p†ˆ km kn 1 ‡ n2p m …1 p† n1‡n2 mm ˆn1 n 2 n 1 ‡ n2; k ˆ 0; 1; ...; min…n 1 ; m†; …6:12†k m k mwhere we have used the result given in Example 6.3 that X ‡ Y is binomiallydistributed with parameters (n 1 ‡ n 2 ,p).The distribution given by Equation (6.12) is known as the hypergeometricdistribution. It arises as distributions in such cases as the number of black ballsthat are chosen when a sample of m balls is randomly selected from a lot ofn items having n 1 black balls and n 2 white balls ( n 1 ‡ n 2 ˆ n). Let randomvariable Z be this number. We have, from Equation (6.12), on replacing n 2by n n 1 , p Z …k† ˆn1 n n1 n; k ˆ 0; 1; ...; min…n 1 ; m†: …6:13†k m k m6.1.2 GEOMETRIC DISTRIBUTIONAnother 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 thisnumber, it is a discrete random variable with possible integer values rangingfrom one to infinity. Its pmf is easily computed to bep X …k† ˆP…FF |‚‚‚‚{z‚‚‚‚} ...F S† ˆP …F†P…F† ...P…F† P…S†|‚‚‚‚‚‚‚‚‚‚‚‚‚{z‚‚‚‚‚‚‚‚‚‚‚‚‚}k 1k 1ˆ q k 1 p; k ˆ 1; 2; ...:…6:14†This distribution is known as the geometric distribution with parameter p,where the name stems from its similarity to the familiar terms in geometricprogression. A plot of p X (k) is given in Figure 6.1.TLFeBOOK