Isye 2027

# In the same way the sum of r independent exponential

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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...
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