Later in this section it is shown that the pdf

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

Ask a homework question - tutors are online