My homework is asking me for different probability distributions given $X \sim \text{Poisson}(\lambda)$
I assume it just wants an equation considering it doesn't give me an actual rate. It first asks me $\text{Pois}(X=2)$, easy enough just plugged in $2$ to the equation correct? The next part asks me $\text{Pois}(X>2)$, in which I compute as $1 - \text{Pois}(X\leq 2)$.
The last part is asking for the distrubtion of $\text{Pois}\left(X^2>2\right)$, which is throwing me off. I know how to calculate second moments etc. but I assume that's only relevant for expectation. Could anyone shed some light and or varify if my methods thus far are correct?
Ian's answer from the comments is completely correct. But it might be beneficial for you to see how we do these problems "in general".
$P(X^2 > 2) = P(X > \sqrt{2} \cup X < -\sqrt{2}) = P(X > \sqrt{2}) + P(X < -\sqrt{2})$
But since $X$ is poisson, and can't take negative values, the second probability is 0.
$P(X^2 > 2) = P(X > \sqrt{2}) = 1 - P(X < 1.414...) = 1 - P(X \leq 1) = 1 - P(X=0) - P(X=1)$