Question

Let \(X\) and \(Y\) be independent random variables with means \(\mu_{x}\) and \(\mu_{y}\) and variances \(\sigma_{x}^{2}\) and \(\sigma_{y}^{2}\). Show that $$ \operatorname{Var}(X Y)=\sigma_{x}^{2} \sigma_{y}^{2}+\mu_{y}^{2} \sigma_{x}^{2}+\mu_{x}^{2} \sigma_{y}^{2} $$

Short answer

To show that \( Var(XY) = \sigma_{x}^{2} \sigma_{y}^{2} + \mu_{y}^{2} \sigma_{x}^{2} + \mu_{x}^{2} \sigma_{y}^{2} \), we first find \( E[XY] \) using the independence of X and Y: \( E[XY] = \mu_{x} \mu_{y} \). Then, we find the second moment \( E[(XY)^{2}] = (\sigma_{x}^{2}+\mu_{x}^{2})(\sigma_{y}^{2}+\mu_{y}^{2}) \). Using the variance formula, \( Var(XY) = E[(XY)^{2}] - (E[XY])^{2} \), and simplifying, we arrive at the desired result.

Step-by-step solution

Find the expectation \( E[XY] \)

Since X and Y are independent, we can use the property \( E[XY]=E[X]E[Y] \) for independent random variables: \( E[XY] = E[X]E[Y] = \mu_{x} \mu_{y} \)

Find the second moment \( E[(XY)^{2}] \)

Again, using the independence of X and Y, we can find the second moment of XY: \( E[(XY)^{2}] = E[X^{2} Y^{2}] = E[X^{2}]E[Y^{2}] \) Now, let's find \( E[X^{2}] \) and \( E[Y^{2}] \) separately: \( E[X^{2}] = Var(X) + (E[X])^{2} = \sigma_{x}^{2} + \mu_{x}^{2} \) \( E[Y^{2}] = Var(Y) + (E[Y])^{2} = \sigma_{y}^{2} + \mu_{y}^{2} \) Now, substitute these back into the formula for \( E[(XY)^{2}] \): \( E[(XY)^{2}] = (\sigma_{x}^{2}+\mu_{x}^{2})(\sigma_{y}^{2}+\mu_{y}^{2}) \)

Calculate the variance of XY using the variance formula

The variance formula is: \( Var(XY) = E[(XY)^{2}] - (E[XY])^{2} \) Plug in the expressions we found for \( E[XY] \) and \( E[(XY)^{2}] \): \( Var(XY) = (\sigma_{x}^{2}+\mu_{x}^{2})(\sigma_{y}^{2}+\mu_{y}^{2}) - (\mu_{x} \mu_{y})^{2} \)

Simplify the expression to get the desired result

Now, expand the expression and simplify: \( Var(XY) = \sigma_{x}^{2} \sigma_{y}^{2} + \mu_{y}^{2} \sigma_{x}^{2} + \mu_{x}^{2} \sigma_{y}^{2} + \mu_{x}^{2} \mu_{y}^{2} - \mu_{x}^{2} \mu_{y}^{2} \) The terms with \( \mu_{x}^{2} \mu_{y}^{2} \) cancel each other out: \( Var(XY) = \sigma_{x}^{2} \sigma_{y}^{2} + \mu_{y}^{2} \sigma_{x}^{2} + \mu_{x}^{2} \sigma_{y}^{2} \) Thus, we have shown the desired result.