Poisson Probability Generating Function

For a non-negative integer-valued random variable $X$, the **probability generating function** (PGF) is defined as $p_X(z) = E[z^X], \quad |z| \leq 1.$ Derive the PGF for $X \sim \text{Poisson}(\lambda)$. Then, with $\lambda = 2$, compute $p_X(1/2)$ to the nearest thousandth.