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    Variance of product of two random variables




    Hi,

    I want to find the variance of the product of two random variables normally distributed with common mean and variance.

    X, Y normally distributed with common mean and variance Var(X) = Var(Y); X, Y are independent identical distributed.

    what is variance(x * y)?

    For the special case where the mean is 0 & non zero Cov(x, x)=0, after running some calculations on a list of random numbers it appears to follow the pattern:
    Var(x*y) = Var(X)^2 = Var(Y)^2

    For the special case where the mean is 0 and where x=x for each realization of x {non zero Cov(x, x)}, after running some calculations on a such list of random numbers it appears to follow the pattern:
    Var(x*x) = 2*Var(X)^2 = 2*Var(Y)^2.

    To make this case clear say the random number stream is

    2.106, -1.392, 4.435, 1.940, -2.933, ...

    then I am calculating the variance of: 2.106*2.106, (-1.392*-1.392), (4.435*4.435), ...

    Confirmation of this apparent relationship would help, a general formula would be better, a proof or location of a proof would be much better
    Thank you Peter
    Last edited by PeterVincent; 10-04-2008 at 07:05 AM. Reason: Forgot to include that X, Y are independent identical distributed random variables

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    Quote Originally Posted by PeterVincent View Post
    Hi,

    I want to find the variance of the product of two random variables normally distributed with common mean and variance.

    X, Y normally distributed with common mean and variance Var(X) = Var(Y)

    what is variance(x * y)?

    Thank you Peter

    I would suggest a starting point for setting up a proof might be (assuming that you're implying that X and Y can be correlated):

    Let Z, E1, and E2 be independent standard normal random variables. Let X and Y be linear combinations of these variables as follows:

    X = r*Z + Sqrt[1-r^2]*E1

    and

    Y= r*Z + Sqrt[1-r^2]*E2.

    where 0<r<1.

    Now, determine the variance of X*Y by taking expectations (and using the definition of variance) of the product of the expansion of the right-hand sides of X and Y. Hint: the correlation between X and Y will be r^2.

    Generalize your result.
    Last edited by Dragan; 10-04-2008 at 02:52 AM.

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    Variance of product of two independent random variables


    Dragan,
    Sorry for wasting your time.

    I should have stated that X, Y are independent identical distributed.

    I assumed that I had stated it and never checked my submission.

    Peter

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