I have two questions regarding maximising the following function defined for $x, y \geq 0$:
$f(x, y) = \displaystyle \sum_{i = 0}^{\infty} \frac{x^i e^{-x}}{i!} \frac{y^i e^{-y}}{i!}$ when $x, y > 0$,
$f(0, 0) = 1$,
$f(0, y) = e^{-y}$, and
$f(x, 0) = e^{-x}$
1) Is the value $f(x, x)$ monotonically decreasing with respect to increasing $x$?
2) For a fixed $x$, is the maximum achieved when $y = x$?