Suposse that $\{N_t\}$ is a Poisson process with rate $\lambda>0$, and the arrival times are $S_1,S_2,\dots$. Evaluate the following in terms of $\lambda$.
$(i) \, \mathbb{P}(N_1 \geq 1, N_3 \leq 2)$
$(ii) \, \mathbb{P}(S_1 > 1, S_2-S_1 \leq 1/2)$
The idea I have is to try to make disjoint partitions of the time intervals in order to use the property of independent increments. But I'm having trouble doing it.
$$\mathbb P\left[N_1 \ge 1 \cap N_3 \le 2\right] = \mathbb P\left[N_1 \ge 1\cap N_1 \le 2 \cap N_3 - N_1 \le 2 - N_1\right] = \mathbb P\left[N_1 = 1\right] \mathbb P\left[N_3 - N_1 \le 1\right] + \mathbb P\left[N_1 = 2\right] \mathbb P\left[N_3 - N_1 =0\right] = \frac{\lambda}{1!}\left(1 + \frac{\lambda}{1!}\right)e^{-3\lambda} + \frac{\lambda^2}{2!}e^{-3\lambda} = \lambda\left(1 + \frac{3}{2}\lambda\right)e^{-3\lambda}$$
$$\mathbb P\left[S_1 > 1 \cap S_2 - S_1 \le \frac12\right] = e^{-\lambda}\left(1 - e^{-\frac\lambda2}\right)$$