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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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 Spring '08
 Zahrn
 The Land

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