# Thread: Difference of Two Squared Normal Random Variables

1. ## 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.

2. Originally Posted by siffi28
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

3. 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?