Question

Question: Show that if\(X\) and \(Y\) are independent random variables, then \(V\left( {XY} \right) = E{\left( X \right)^2}V\left( Y \right) + E{\left( Y \right)^2}V\left( X \right) + V\left( X \right)V\left( Y \right)\)

Short answer

Answer

The statement \(V\left( {XY} \right) = E{\left( X \right)^2}V\left( Y \right) + E{\left( Y \right)^2}V\left( X \right) + V\left( X \right)V\left( Y \right)\) is proved if \(X\) and \(Y\) are independent random variables.

Step-by-step solution

Given information

\(X\) and \(Y\) are independent random variables.

Definition and formula used

A random variable is a mathematical formalization of a quantity or object which depends on random events.

Calculation  

Simplify the right hand side by substitution.

\(\begin{aligned}{c}E{\left( X \right)^2}V\left( Y \right) + E{\left( Y \right)^2}V\left( X \right) + V\left( X \right)V\left( Y \right) &= E{\left( X \right)^2}\left( {E\left( {{Y^2}} \right) - E{{\left( Y \right)}^2}} \right) + E{\left( Y \right)^2}\left( {E\left( {{X^2}} \right) - E{{\left( X \right)}^2}} \right) + \\\left( {E\left( {{X^2}} \right) - E{{\left( X \right)}^2}} \right)\left( {E\left( {{Y^2}} \right) - E{{\left( Y \right)}^2}} \right)\\ &= E{\left( X \right)^2}E\left( {{Y^2}} \right) - E{\left( X \right)^2}E{\left( Y \right)^2} + E{\left( Y \right)^2}E\left( {{X^2}} \right) - E{\left( Y \right)^2}E{\left( X \right)^2}\\ + E\left( {{X^2}} \right)E\left( {{Y^2}} \right) - E\left( {{X^2}} \right)E{\left( Y \right)^2} - E{\left( X \right)^2}E\left( {{Y^2}} \right) - E{\left( X \right)^2}E{\left( Y \right)^2}\\ &= E\left( {{X^2}} \right)E\left( {{Y^2}} \right) - E{\left( X \right)^2}E{\left( Y \right)^2}\\ &= E\left( {{{\left( {XY} \right)}^2}} \right) - E{\left( {XY} \right)^2}\\ &= Var\left( {xy} \right)\end{aligned}\)

Hence proved.