Find the law of random variable

Question

Let $Y$ have the Poisson distribution $Y \sim P(\lambda )$ and let $Z = \left\{ \begin{gathered} \frac{Y}{2}{\text{ if }}Y{\text{ is even}} \hfill \\ \frac{{\left( {1 - Y} \right)}}{2}{\text{ if }}Y{\text{ is odd}} \hfill \\ \end{gathered} \right.$

How can i find the law of $Z$ ?

Answer 1

Let $p_Y$, the Poisson distribution

For $k<0$, $P(Z=k)=p_Y(1-2k)$ For $k=0$, $P(Z=0)=p_Y(0)$ For $k>0$, $P(Z=k)=p_Y(2k)$