Given a sequence $Y_i$ of i.i.d random variables with finite mean, representing the interarrival times. Define $S_n = \sum_{i=1}^n Y_i$ to be the arrival times, and $N_t = \sum_{n=1}^\infty \bf{1}_{\{S_n \le t\}}$ denoting the number of arrivals in the interval $[0,t]$.
Prove that $E(N_t) = \frac{t}{E(Y_1)}$
It seems like this should be immediate from the definitions, but I can't seem to prove it.
A simple counter example: take $Y_i=1$ for all $i$. This leads to $E(N_t)=⌊t⌋$.