The numbers of additional trials required for each

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Unformatted text preview: Bernoulli process and the negative binomial distribution Recall that a random variable has the Bernoulli distribution with parameter p if it is equal to one with probability p and to zero otherwise. A Bernoulli process is an infinite sequence, X1 , X2 , . . . , of Bernoulli random variables, all with the same parameter p, and independent of each other. Therefore, for example, P {X5 = 1, X6 = 1, X7 = 0, X12 = 1} = p3 (1 − p). We call Xk the outcome of the k th trial. It is natural to introduce some additional random variables associated with a Bernoulli process, as shown in Figure 2.6. Let L1 be the number of trials needed until the outcome of a trial is one. As seen in Section 2.5, L1 has the geometric distribution with parameter p. Let L2 denote the number of trials, after the first L1 trials, until again the outcome of a trial is one. Then L2 also has the geometric distribution with parameter p. In general, let Lj denote the number of trials needed after the first L1 + · · · + Lj...
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