Variance of product of two random variables

#1
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
 
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Dragan

Super Moderator
#2
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.
 
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#3
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