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. DISCRETETYPE 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...
View
Full
Document
 Spring '08
 Zahrn
 The Land

Click to edit the document details