In the same way the sum of r independent exponential

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: P {N = n} i!j ! T ni p (1 − p)n−i , i i T −τ T j τ where p = T . As a function of i, the answer is thus the pmf of a binomial distribution. This result is indicative of the following stronger property that can be shown: given there are n counts during [0, T ], the times of the counts are independent and uniformly distributed over the interval [0, T ]. 84 CHAPTER 3. CONTINUOUS-TYPE RANDOM VARIABLES (e) Since N1 and N2 are numbers of counts in disjoint time intervals, they are independent. Therefore, P (N = n|N1 = i) = P (N2 = j |N1 = i) = P {N2 = j } = e−λ(T −τ ) (λ(T − τ ))j . j! −λ(T −τ ) n−i (λ(T −τ )) . That is, given N1 = i, the which can also be written as P (N = n|N1 = i) = e (n−i)! total number of counts is i plus a random number of counts. The random number of counts has the Poisson distribution with mean λ(T − τ ). Example 3.5.4 Calls arrive to a cell in a certain wireless communication system according to a Poisson process with arrival rate λ = 2 cal...
View Full Document

Ask a homework question - tutors are online