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Math Help - Difference of Two Squared Normal Random Variables

  1. #1
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    Difference of Two Squared Normal Random Variables

    Hello everyone,

    If X1 and X2 are two normal random variables, then assuming

    X = (X1)^2 + (X2)^2

    X is Chi-Squared random variable with two degrees of freedom. My question is that if

    Y = (X1)^2 - (X2)^2

    then what is the distribution function of Y and what is its mean and variance?

    Any help will be greatly appreciated.
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  2. #2
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    Quote Originally Posted by siffi28 View Post
    Hello everyone,

    If X1 and X2 are two normal random variables, then assuming

    X = (X1)^2 + (X2)^2

    X is Chi-Squared random variable with two degrees of freedom. My question is that if

    Y = (X1)^2 - (X2)^2

    then what is the distribution function of Y and what is its mean and variance?

    Any help will be greatly appreciated.
    Y = (X_1 + X_2) (X_1 - X_2) and since the sum and the difference of two independent normal random variables is also a normal random variable you should think about what the distribution of the product of two normal random variables is. A clue might be found here: Normal distribution - Wikipedia, the free encyclopedia
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  3. #3
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    Thank you very much for the quick reply.

    So the product of two independent normal random variables has a pdf given by modified bessel function of second kind.

    How can I find the variance of this product in terms of the means and variances of the original random variables?
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