An operator receives on the average 20 calls per hour with a Poisson process.
(i) What is the probability that she waits more than 12 minutes before receiving the fifth call ?
Let $X$ be the waiting time to receive exactly 2 calls.?
(ii) Find the p.d.f. for $X$.
(iii) Find the moment generating function for X and use it to calculate $E(X)$ and $Var(X)$.
I only know the formula for the probability that she waits for more than 12 minutes before recovering the first call which is $\int_5^\infty 1/3e^{-x/3}\mathrm d~x$ but I don't know how to do it with fifth call...
Thanks!!
(I) The probability that she waits more than twelve minutes before receiving the fifth call is the probability that less than five calls occur within the twelve minutes. The count of calls received in the interval is Poisson distributed.
(II) Simlarly determine the probability that she waits more than $t$ (in minutes) before receiving the second call, subtract that from one to find the CDF, and differentiate to find the pdf.
(III) Using that pdf, find the expected value of $e^{tX}$, then use the Taylor series expansion of the moment generating function to find the first and second moments. Use them to find what you require.
$$M_t(X)~=~\mathsf E(e^{tX}) ~=~ \sum\limits_{n=0}^\infty \dfrac{t^n\,\mathsf E(X^n)}{n!}$$