# Thread: Bivariate random variable problem

1. ## Bivariate random variable problem

Hello

I am trying to solve this problem:

A coin is given with probability 1/3 for head (H) and 2/3 for tail (T).
The coin is being drawn N times, where N is a Poisson random variable with E(N)=1. The drawing of the coin and N are independent. Let X be the number of heads (H) in the N draws. What is the correlation coefficient of X and N ?

So I started this by creating a table as if it was a finite problem, just to see how it behaves, but it didn't lead me too far. Since there is independence, every event P(X=x , N=n) is equal to P(X=x|N=n)*P(N=n). So this is like a tree diagram sample space. In order to find the correlation, I need the covariance and the variances. The variance of N, it's easy, 1. How do I find the rest of the stuff ?

Thanks !

2. ## Re: Bivariate random variable problem

Firstly note that has a compound distribution, i.e. it can be expressed as

with the convention that when

are i.i.d. and independent of

Next you can use Law of total variance and covariance to help you.

http://en.wikipedia.org/wiki/Law_of_total_variance
http://en.wikipedia.org/wiki/Law_of_total_covariance

Let's try first. (You will need to know some basic facts/properties.)

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