What is the standard deviation of the product of two random variables that each have Gaussian Distributions? I don't even know where to begin on this problem.
[Math] Standard deviation of the product of gaussians
standard deviationstatistics
standard deviationstatistics
What is the standard deviation of the product of two random variables that each have Gaussian Distributions? I don't even know where to begin on this problem.
Best Answer
Let $X$ and $Y$ be independent random variables each with normal distribution (the means and variances need not be the same). Let $W=XY$. Then $$\text{Var}(W)=E((XY)^2)-(E(XY))^2.$$ We need to compute the two expectations on the right.
By independence, $E(XY)=E(X)E(Y)$. Also, $E(X^2Y^2)=E(X^2)E(Y^2)$.
Since $E(X^2)=\text{Var}(X) +(E(X))^2$, with a similar expression for $E(Y^2)$, once we know the mean and variance of $X$ and $Y$, we can use the above equations to find $\text{Var}(XY)$.
Added: Let the means be $\mu$ and $\nu$, and the variances be $\sigma^2$ and $\tau^2$. Then the variance of $XY$ is, by the above argument, equal to $$(\sigma^2+\mu^2)(\tau^2+\nu^2)-\mu^2\nu^2.$$ This simplifies to $\sigma^2\tau^2+\sigma^2\nu^2+\tau^2\mu^2$.
Note that we did not use the normality of $X$ and $Y$.