$(4)(2$ Points$)$ Let $x$ be a $Z-$valued random variable. Show that $x$ is Poisson

distributed with parameter $=1$ if and only if for any bounded function

$f$:$ZR$ one has

$E(f(x+1))=E(xf(x))$

Hint: For the reverse part you might want to try the explicit function $f=I_{\{a+1\}}$

for some ainZ, start with $a=-1$ and define the constant $c$:$=P(Y=0).$ Then

express $P(Y=a)$ in terms of $P(Y=0)$ if $a0$ and in terms of $P(Y=-1)$ if