Calculating the mass function of maximum of a sum

Question

Find an expression for the mass function of $N(t)$ in a renewal process whose interarrival times $X_i$ are a) poisson distributed with paramter $\lambda$ and b) gamma distributed $\Gamma(\lambda,b)$.

As a recall: $N(t)=\max\{ n: T_n \leq t \}$ with $T_n=X_1+...+X_n.$

I hope anybody can help me a little bit in a) so that I can do part b) by myself.

a)

$$f_{N(t)}(x)=\Pr(N(t)=x)=\Pr(\max\{ n: T_n \leq t \}=x)=\Pr(T_k )=f_{T_k},$$ for all $k\leq x$. Is that right?

Answer 1

Hint: Let $T_0=0$, then, for every $n\geqslant0$, $$P(N(t)=n)=P(T_n\leqslant t)-P(T_{n+1}\leqslant t).$$