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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?

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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

$$ \frac{-k\cdot\frac1n\cdot\frac1n}{k\frac1n\left(1-\frac1n\right)}=-\frac1{n-1}\;. $$

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    $\begingroup$ I am very grateful! Thank you :) $\endgroup$
    – wiwnes691
    Feb 26, 2020 at 21:12

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