

13th April 2011, 21:13 #1
Member level 4
variance of the product of two random variables
If I have two RV's X and Y (which are not necessarily independent), what would be the approach to find the variance of their product assuming that I know variances of X and Y, i.e. if Z=XY, what is var(Z)?
I searched on google and found some suggestions, but most of them were based on the assumption that X and Y are independent. What would be a more general solution?
Thanks a lot.

13th April 2011, 21:13

14th April 2011, 13:33 #2
Member level 1
Re: variance of the product of two random variables
If there exists corelation between X and Y, then finding the variance of Z becomes slightly more complicated due to the requirement of knowing the covariance of X and Y.
Syntax: [] = covariance, <> = variance, "" = Mean, () = Normal Paranthese
<Z> = [XY] . ( [XY] + 2."X"."Y" ) + <X>.<Y> + <X>("Y"^2) + <Y>("X"^2)
Hope it helps.
{Btw, it can be simplified into just 3 terms: <Z> = "XY"^2 + ("XX"."YY")  (2 . ("X"^2) . ("Y"^2)), but as I said you need to take covariance into account. }

14th April 2011, 13:33

15th April 2011, 14:35 #3
Member level 3
Re: variance of the product of two random variables
If you want to develop considerable knowledge, read functions of random variables from Papoulius book.