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Thread: Bivariate random variable problem

  1. #1
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    Bivariate random variable problem


    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. #2
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    Re: Bivariate random variable problem

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

    X = \sum_{i=1}^N Z_i

    with the convention that X = 0 when N = 0

    Z_i are i.i.d. \text{Bernoulli}\left(\frac {1} {3} \right) and independent of N \sim \text{Poisson}(1)

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


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

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