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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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 Spring '08
 Zahrn
 The Land

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