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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?

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    $X^2>2$ is the same as $X>1$, since $X$ is nonnegative integer valued. – 2017-02-14
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    So all you did was square root the X^2 and 2? Right, thought of that but was confused with the 2^(1/2). So you just rounded it? Or would it be more accurate to say you took the floor of the number. Such as if X^2 > 3, would that = X > 2 or X > 1 – 2017-02-14

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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.

$P(X^2 > 2) = P(X > \sqrt{2}) = 1 - P(X < 1.414...) = 1 - P(X \leq 1) = 1 - P(X=0) - P(X=1)$

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    I understand now, thanks you two! – 2017-02-14