Poisson distribution. $y = 5 (\epsilon + 1)^{-1}$ probability that $y$ is integer

Question

I have ran into trouble that I have no idea how to solve.

So the problem is:

We have random variable $\epsilon$ which is distributed by Poisson distribution with parameter $\lambda$ = 0.46. Then $$y = 5 (\epsilon + 1)^{-1}$$ What is the probability that $y$ is integer?

Answer 1

A Poisson random variable can only take values $\left\{0,1,2,3,4,\dotsc\right\}$, therefore, we need to find the values of $n$ such that $\frac{5}{n+1}\in\mathbb{Z}$. This is only true when $n+1=1$ or $n+1=5$. Therefore we get that $n=0$ or $n=4$. Now simply, \begin{align*} \mathbb{P}_{\epsilon}\left(n=0\text{ or }4\right)&=\mathrm{e}^{-0.46}\left(\frac{0.46^0}{0!}+\frac{0.46^4}{4!}\right) \\ &=0.631284+0.001178 \\ &= 0.632462 \end{align*}