Introduction to Probability by George G. Roussas

Author:George G. Roussas
Language: eng
Format: epub, pdf
ISBN: 9780128001981
Publisher: Elsevier Science
Published: 2013-11-30T05:00:00+00:00

(9.11)

where are integers with , and, of course, . The distribution given by (9.11) is the multinomial with parameters n and , and the r.v.’s are said to have (jointly) the multinomial distribution with these parameters.

That the right-hand side of (9.11) is the right formula for the joint probabilities ensues as follows: By independence, the probability that occurs times, , in specified positions, is given by: regardless of the positions of occurrence of ’s. The different ways of choosing the positions for the occurrence of , is equal to: (by Theorem 1 in Chapter 2). Writing out each term in factorial form and making the obvious cancellations, we arrive at: (see also Exercise 2.1).

For illustrative purposes, let us consider the following example.

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