Problem 1 A local policeman drives from intersection to inte

Problem 1: A local policeman drives from intersection to intersection in time X, which is exponentially distributed with parameter lambda. At each intersection, the policeman observes and reports a car accident with probability p. (The activity does not slow his driving at all). Independently of all else, the policemen received extremely brief radio calls in a Poisson manner with an average rate of t calls an hour. i) Find the PMF for N, the number of intersections the policemen visits up to and including the first one where he reports the first accident. ii) Find the PDF for q, the length of time the policeman drives between reporting accidents iii) What is the PMF for M, the number of accidents which the policeman reports in two hours? iv) What is the PMF for K, the number of accidents the policeman reports between receiving two successive radio calls?

Solution

Intersection to intersection in time x : exponential with lembda

Car accident notices - prob =p

no of radio calls : mu Poisson

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PMF of N = Take no of interesection he visits per hour = 1/x

Hence Poisson distribution with 1/x for 1 hour as parameter

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PDF of q = Binomial distributin with (N,q) as parameters.

iii) M, no of accidents Poisson with Np as parameter

iv) exponential distribution