Q:

Let $\{X(t); t\ge 0\}$ be a Poisson process with rate $\lambda =2$. Find the probability $\Pr\{X(1)\le 2\}$.

Let $\{X(t); t\ge 0\}$ be a Poisson process with rate $\lambda =2$. Find the probability $\Pr\{X(1)\le 2\}$.

I don't know how to calculate this kind of probabilities, I just know how to calculate when it has an equal sign $=$ but not with the $\leq$ sign.
Could someone help me?

A:

Hint: Loosely speaking, $X(t)$ counts the number of "arrivals" until time $t$. If you know how to "calculate when it has an equal sign", then notice that
$$P(X(1)\leq 2) = P\left(\bigcup_{k =0}^2 X(1) = k\right).$$

