Unformatted text preview: on with
parameters k and p. More generally, for 0 ≤ k < l, the increment Cl − Ck is the number of
ones in k − l Bernoulli trials, so it has the binomial distribution with parameters l − k and p.
Also, the increments of C over nonoverlapping intervals are independent.
• The cumulative numbers of trials for j ones, for j ≥ 0: (S0 , S1 , S2 , . . .). As discussed below,
for integers r ≥ 1, Sr has the negative binomial distribution with parameters r and p. Negative binomial distribution In the remainder of this section we discuss the distribution
of Sr , which is the number of trials required for r ones, for r ≥ 1. The possible values of Sr are
r, r + 1, r + 2, . . . . So let n ≥ r, and let k = n − r. The event {Sr = n} is determined by the outcomes
of the ﬁrst n trials. The event is true if and only if there are r − 1 ones and k zeros in the ﬁrst
−1
k + r − 1 trials, and trial n is a one. There are n−1 such sequences of length n, and each has
r
probability pr (1 − p)...
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 Spring '08
 Zahrn
 The Land

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