Let i j n be nonnegative integers such that n i j

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Unformatted text preview: 0. That is, as h → 0, the complementary CDF of T converges to the complementary CDF of an exponentially distributed random variable with parameter λ. So also the CDF of T converges to the CDF of an exponentially distributed random variable with parameter λ. In summary, the CDF of h times a geometrically distributed random variable with parameter p = hλ converges to the CDF of an exponential random variable with parameter λ. 3.5 Poisson processes Bernoulli processes are discussed in Section 2.6. Here we examine Poisson processes. Just as exponential random variables are limits of scaled geometric random variables (as seen in Section 3.4), Poisson processes are limits of scaled Bernoulli processes. 3.5.1 Time-scaled Bernoulli processes Let X1 , X2 , . . . form a Bernoulli process with parameter p, with 0 ≤ p ≤ 1. As discussed in Section 2.6, this means that the random variables are independent, and P {Xk = 1} = p and P {Xk = 0} = 1 − p. We say that the k th trial results in a count if Xk = 1. Let h > 0, with h representing 3.5. POISSON PROCES...
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This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Tech.

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