Erlang Distribution as Poisson Process Waiting Time

Arrivals follow a Poisson process with rate $\lambda > 0$. Let $T_k$ denote the time of the $k$-th arrival.

(a) Using the relationship $P(T_k > t) = P(N(t) < k)$ where $N(t) \sim \text{Poisson}(\lambda t)$, write the CDF of $T_k$ and differentiate to obtain its PDF.

(b) Identify the distribution by name and state its mean and variance.

(c) For $\lambda = 1$ and $k = 3$, compute $P(T_3 > 2)$ as an exact numerical value.