Unformatted text preview: −1 trials, until again the outcome of a trial is one. The 2.6. BERNOULLI PROCESS AND THE NEGATIVE BINOMIAL DISTRIBUTION 39 C’s 3
2 X’s 1 k
S2 30 40 50 L3 S3 Figure 2.6: A Bernoulli process.
random variables L1 , L2 , . . . are independent random variables, each geometrically distributed with
parameter p. Note that the L’s are determined by the X ’s. The converse is also true; the values of
all the L’s determine the values of all the X ’s.
We shall give two more ways to describe the process. Let Sj denote the total number of
trials, counting from the very ﬁrst one, until a total of j trials have outcome one. Equivalently,
Sj = L1 + L2 + · · · Lj , for j ≥ 1. The L’s determine the S ’s, and the converse is also true:
Lj = Sj − Sj −1 for j ≥ 1, with the understanding S0 = 0.
Let Ck denote the cumulative number of ones in the ﬁrst k trials. That is, Ck = X1 + X2 + · · · +
Xk . By convention, C0 = 0. The sequence (Ck : k ≥ 0) is sometimes called the counting se...
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