It is shown in example 481 that varsr rvarl1 r1p

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Unformatted text preview: quence of the Bernoulli process, because it counts the number of ones vs. the number of trials. Clearly, the counting sequence is determined by the X ’s. Conversely, the X ’s are determined by the counting sequence: Xk = Ck − Ck−1 for k ≥ 0. If 0 ≤ k ≤ l, the difference Ck − Cl is called the increment of C over the interval (k, l] = {k + 1, k + 2, · · · , l}. It is the number of trials in the interval with outcome equal to one. 40 CHAPTER 2. DISCRETE-TYPE RANDOM VARIABLES To summarize, there are four ways to describe the same random sequence: • The underlying Bernoulli sequence (X1 , X2 , . . .). The random variables X1 , X2 , · · · are independent Bernoulli random variables with parameter p. • The numbers of additional trials required for each successive one to be observed: L1 , L2 , · · · . The random variables L1 , L2 , · · · are independent, geometrically distributed random variables with parameter p. • The cumulative number of ones in k trials, for k ≥ 0, (C0 , C1 , C2 , . . .). For k fixed, Ck is the number of ones in k independent Bernoulli trials, so it has the binomial distributi...
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