# Correlation between dependent variables (binomial distribution)

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?

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

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