Let $T_i$ (i = 0, 1, 2,...) be the arrival times of a Poisson process of rate $\lambda$. Let $$ N = inf\{k > 1:T_k - T_{k-1} > T_1\}$$ Find E(N).
I understand that the solution, which is first to calculate P(N $\geq$ n) = $\frac{1}{n-1}$ and then do the summation. But I am wondering why cannot I regard N as a Geometric distribution with success probability $p = P(T_k - T_{k-1} > T_1)$ and then E(N) = $\frac{1}{p}$?
Thanks.