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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?

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    $\begingroup$ Do you know multinomial distribution? $\endgroup$
    – Kroki
    Jun 2 at 17:21
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    $\begingroup$ Are $X_1, \ldots, X_n$ independent? $\endgroup$
    – balddraz
    Jun 2 at 17:22
  • $\begingroup$ Sorry, I forgot to add that $X_1,...,X_n$ are independent. Thank you for pointing it out. $\endgroup$
    – martjian
    Jun 2 at 17:27
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    $\begingroup$ @Youem, now that you mentioned it.... I'll give it a shot. Thanks! $\endgroup$
    – martjian
    Jun 2 at 17:34

1 Answer 1

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The random vector $\left(Z_1,\ldots,Z_n\right)$ has a multinomial distribution such that $p_i = \frac1n$. So for $i\neq j$, $$\text{Cov}\left[X_i, X_j\right] = -kp_ip_j = -\frac k{n^2}$$

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