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Thread: Joint Distributions and a 3rd Variable

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    Exclamation Joint Distributions and a 3rd Variable




    I am struggling with this probability problem.

    Consider the random variables X and Y with joint distribution as given below.
    X
    0 1
    Y 1 .2 .2
    2 .5 .1
    1

    Calculate the probability mass function of the new random variable Z≡X+Y

    z___|____|____|______|_____|
    p(z)_|____|____|______|_____|



    Calculate the variance of Z, V[Z]≡σ_Z^2 and show that V[Z]=V[X]+V[Y]+2cov[X,Y].

    I am not sure how to complete the table. I think that I may be able to work on the variances on my own.

  2. #2
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    Re: Joint Distributions and a 3rd Variable


    So in the given joint pmf table, you see there is 4 support points (possible cases) for (X, Y). Try to see what value does X + Y take in each case and you fill the table accordingly. As you may expected, sometimes you may have two or more different cases having the identical value of X + Y, which you will need to sum the probabilities of those cases for the pmf of Z.

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