Let $\xi$ and $\mu$ be independent random variables with the same Poisson distribution ($\lambda$ as the parameter). For any given natural $k\ge2$ consider random vector $(\gamma, \delta)$ with density given by $$P(\gamma = a, \delta = b) = P(\xi = a,\ \ \mu = b \ \ |\ \xi + \mu =k)$$
Find the correlation of $\gamma$ and $ \delta$.
Attempt: using the definition of conditional probability one can obtain that given $k = a + b$ $$P(\gamma = a, \delta = b)= \frac{P(\xi = a,\ \ \mu = k-b)}{P( \xi +\mu = k)}$$
Now from the fact that sum of Poission random variables is Poission random variable with the parameter equal of the sum of parameters of summed variables
$$P(\gamma = a, \delta = b)= \binom{a+b}{a} \frac{1}{2^{a+b}}$$
I approach the covariance with formula $cov(\gamma,\delta) = E(\gamma \delta)-E(\gamma)E(\delta)$. Trying to compute $E(\gamma \delta)$. If I am not wrong, it would be $\sum_{a\in\mathbb{N}} \sum_{b\in\mathbb{N}}ab\binom{a+b}{a} \frac{1}{2^{a+b}}$. But the series is divergent, therefore $E(\gamma \delta)$ is not a number.
Is it correct? I suspect that I misinterpret formula $E(f(n)) = \sum f(n)P(n)$ when trying to compute $E(\gamma \delta)$.