People arrive at random times and independently at a bus stop and wait for the bus to arrive. The bus arrives at this stop once every hour. Thus, the waiting times of the people follow a uniform distribution over the unit interval, i.e., $T_1, T_2,\ldots\sim U(0,1)$. We are interested in the minimum number of people that will have a combined waiting time that exceeds $45$ minutes i.e., $N=\min_{k\geq 1}\left(\{X_1+X_2+\dots+X_k > 3/4\} \cap \{X_1+X_2+\dots+X_{k-1} \leq 3/4\}\right). $ a) Show by induction that $ P(N=k) = \frac{(3/4)^{k-1}}{(k-1)!}-\frac{(3/4)^k}{k!},\quad k = 1,2,\dots $
b) Find the mean and variance of $N$.
I got stuck on the induction part of the problem and I got the a different answer to the mean and variance of $N$ than the book's. Also I apologize for the format of the problem. I am new to this website and do not know how to use latex very well. Can someone please help? Thank you for your time and consideration!