Number of buses stopping till time $t$ 
Let $\displaystyle \left(X_n\right)_{n\geq 1}$ be i.i.d random variables on $\displaystyle \left(\Omega,\mathcal A,\mathbb P\right)$, $X_1$ being an exponential random variable with parameter $1$. Define $T_0=0$, $T_n=X_1+\ldots+X_n$, and for any $t>0$,
$$\displaystyle N_t=\max\left\{n\geq 0 \mid T_n\leq t\right\}.$$

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*For any $n\geq 1$, calculate the joint distribution of $\displaystyle \left(T_1,\ldots,T_n\right)$.


*Deduce the distribution of $N_t$, for arbitrary $t$.


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*Consider 2 arrivals arriving at $T_1$ and $T_2$. The joint pdf $f_{T_1,T_2}(t_1,t_2)=f_{T_1}(t_1)f_{X_2}(t_2-t_1)=\lambda e^{-\lambda t_1}\times \lambda e^{-\lambda (t_2-t_1)}=\lambda^2 e^{-\lambda t_2}$. In a similar vein, you could argue for the $N$-arrivals case that $f_{T_1,T_2,\ldots, T_n}(t_1,t_2,\ldots, t_n)=\lambda^n e^{-\lambda t_n}, 0\le t_1\le t_2\ldots \le t_n$.

*Since this is a Poisson process, $N_t$ is a Poisson random variable with mean $\lambda t$. $P(N_t=n)=\frac{e^{-\lambda t}(\lambda t)^n}{n!}$, $n=0,1,\ldots,$

