Later in this section it is shown that the pdf

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Unformatted text preview: ables from the pdfs of the random variables being added together. But the following method is less work. Notice that for a fixed time t, the event {Tr > t} can be written as {N (t) ≤ r − 1}, because the rth count happens after time t if and only if the number of counts that happened by time t is less than or equal to r − 1. Therefore, r−1 P {Tr > t} = k=0 exp(−λt)(λt)k . k! The pdf is thus fTr (t) = − dP {Tr > t} dt r −1 = exp(−λt) λ k=0 r −1 = exp(−λt) k=0 r −1 = exp(−λt) k=0 = (λt)k − k! λk+1 tk k! r−1 k=1 r−1 − λk+1 tk − k! k=1 r−2 k=0 kλk tk−1 k! λk tk−1 (k − 1)! λk+1 tk k! exp(−λt)λr tr−1 . (r − 1)! The distribution of Tr is called the gamma distribution with parameters r and λ. The mean of Tr is r λ , because Tr is the sum of r random variables, each with mean 1/λ. It is shown in Example 4.8.1 r that Var(Tr ) = λ2 . Recall from Section 3.4 that the exponential distribution is the limit of a scaled geometric random variable. In the same way, the...
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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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