# Correlation and Poisson distribution

Let $$X_1, \ldots, X_n$$ independent random variables with $$\mathrm{Poiss}\left(\lambda \right)$$ distribution. For $$k \ge 2$$ we define random vector $$\left(Z_1, \ldots, Z_n\right)$$ such that: $$\mathbb{P}\left(Z_1=x_1, \ldots, Z_n=x_n\right) = \mathbb{P}\left(X_1=x_1, \ldots, X_n=x_n | X_1 + \cdots + X_n = k\right)$$.

Calculate $$\operatorname{corr}\left(Z_1,Z_2\right).$$

I spent a lot of time thinking how to solve it, but I failed. Can anyone help me?

Recall that multiple Poisson distributions conditioned on their sum yield a multinomial distribution with the probabilities in the same ratio as the Poisson rates. So you’re looking for the correlation between two components of a multinomial distribution, with equal probabilities $$\frac1n$$. The variance of each component is $$k\frac1n\left(1-\frac1n\right)$$, and the covariance of two components is $$-k\cdot\frac1n\cdot\frac1n$$, so the correlation is
$$\frac{-k\cdot\frac1n\cdot\frac1n}{k\frac1n\left(1-\frac1n\right)}=-\frac1{n-1}\;.$$