Calls are made to a telephone helpline according to a Poisson process at an average rate of 3 calls per hour. The number of questions asked by each caller has a geometric distribution (starting at 1) with mean 2.5.
Find the p.g.f. of the total number of questions asked in a period of three hours and use the p.g.f.to find variance and mean of the number of questions asked in a period of three hours.

Question

Calls are made to a telephone helpline according to a Poisson process at an average rate of 3 calls per hour. The number of questions asked by each caller has a geometric distribution (starting at 1) with mean 2.5.
Find the p.g.f. of the total number of questions asked in a period of three hours and use the p.g.f.to find variance and mean of the number of questions asked in a period of three hours.

amistre64 · Answer

im not adept at trying to work a pgf, but this looks reasonably sane :) http://www.youtube.com/watch?v=JVJOF0apfdo

dumbcow · Answer

i dont know how to find the p.g.f but i think you can get mean/variance by using properties of random variables
Let X be num of calls in 3 hours
Let Y be num of questions asked per caller
Let T be total questions asked in 3 hours

$$T = XY$$
$$E(T) = E(X)*E(Y)$$
$$Var(T) = Var(X)*Var(Y) + Var(X)E(Y)^{2} + E(X)^{2} Var(Y)$$