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- 13th April 2011, 21:13 #1

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## 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

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## 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

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## Re: variance of the product of two random variables

If you want to develop considerable knowledge, read functions of random variables from Papoulius book.

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