Good afternoon. I have the following problem concerning Poisson and binomial distributions.
Let $X_1, ..., X_n, n\geq 2$ be independent random variables, Poisson distributed with a parameter $\lambda >0$. Let $k \geq 2$. Let $(Z_1, ..., Z_n)$ be a random vector with the following distribution:
$P(Z_1=x_1,..., Z_n = x_n) = P(X_1 = x_1, ..., X_n = x_n|X_1+...+X_n=k)$.
Calculate the correlation coefficient between $Z_1$ and $Z_2$.
I have already calculated $P(Z_1 = t) =P(X_1 = t|X_1+...+X_n = k)$ and found that it has binomial distrubution with parameters $(k, \frac{1}{n})$. The same holds true for $P(Z_2 = t) =P(X_2 = t|X_1+...+X_n = k)$.
If I'm right, I should be calculating $E(Z_1 Z_2)$ right now.
Can someone please explain to me how to use the fact that these two variables are dependent to calculate that expected value?