this questions throwing me through the loop a bit, any and all clarification would be great thanks.
let $\lambda > 0$ be a parameter, and X an integer-valued Random Variable with
$$P\{X=z\} = \left\{ \begin{array}{R} \frac{\lambda^{|z|}e^{-\lambda}}{|z|!\cdot 2} \mbox{ if z} \neq 0 \mbox{ integer} \\ e^{-\lambda} \mbox{ if } z=0 \end{array}\right. $$
computer the expectation
so the expecation is given by $$\sum_{z \neq 0}{xp(x)} = \sum_{z \neq 0}{z\frac{\lambda^{|z|}e^{-\lambda}}{|z|!\cdot 2}} = \frac{1}{2e^{\lambda}}\sum_{z \neq 0}{z\frac{\lambda^{|z|}}{|z|!}}$$
and this is where i get lost. upon sneaking a look at the answers i see that $E[X]=0$ but i have no idea why.
You don't actually need to compute the summation explicitly:
We have $EX = 0 P[X=0] + \sum_{z=1}^\infty z P[X=z] + \sum_{z=1}^\infty (-z) P[X=-z]$
Now note that $P[X=z] = P[X=-z]$.