If X ~ $P(\lambda )$ $p(2)=2e^{-2}$ , find $P(X>3), E(X) and V(X)

Question

If X ~$ P(\lambda )$, $p(2)=2e^{-2}$ , find $P(X>3), E(X) and V(X)

I don't understand how to find this things with only p(2)

Answer 1

If $X$ has Poisson distribution with parameter $\lambda$, then its probability mass function, $p$, is given by $$p(k) = e^{-\lambda} \frac{\lambda^k}{k!}.$$ Knowing that $p(2) = 2e^{-2}$ we have $$2e^{-2} = p(2) = e^{-\lambda} \frac{\lambda^2}{2!}$$ and then $\lambda = 2$ (check it). Can you continue from here?