# If X and Y are correlated random variables, what is Var(XY)?

Apr 15, 2016

V(XY)=[E^2(X)*Var(Y)+E^2(Y)*Var(X)]+2[E(X)*E(Y)*Cov(X,Y)] +Var(X)*Var(Y)+C(X,Y)^2

#### Explanation:

To derive let's write what $V a r \left(X Y\right)$:

Var[XY] = E[(XY)^2] – {E[XY]}^2
 Var(XY) = color(red)(E[X^2Y^2]) – color(blue)((E[X]E[Y])^2)

$= \textcolor{red}{E \left[{X}^{2}\right] E \left[{Y}^{2}\right]} + C o v \left({X}^{2} , {Y}^{2}\right)$
$= {\left\{\textcolor{b l u e}{E \left[X\right] E \left[Y\right]} + C o v \left(X , Y\right)\right\}}^{2}$
$= \textcolor{b l u e}{{\left\{E \left[X\right] E \left[Y\right]\right\}}^{2} + 2 \left\{E \left[X\right] E \left[Y\right]\right\}} C o v \left(X , Y\right) + C o {v}^{2} \left(X , Y\right)$

$V a r \left(X Y\right) = \textcolor{red}{E \left[{X}^{2}\right] E \left[{Y}^{2}\right]} + C o v \left({X}^{2} , {Y}^{2}\right) - \left\{\textcolor{b l u e}{{E}^{2} \left[X\right] {E}^{2} \left[Y\right]} + 2 E \left[X\right] E \left[Y\right] C o v \left(X , Y\right) + C o {v}^{2} \left(X , Y\right)\right\}$

Now if $X \mathmr{and} Y$ were independent the covariance will vanish which implies that correlation is also zero. However, in this case your random variables are correlated, thus the covariance stays on the above equation.
Now if you want to take it further you can sub the below for the expectation and variance operators. But i will stop here
E[X]=mu_x; E[Y]=mu_y
$E \left[X Y\right] = E \left[X\right] E \left[Y\right] = {\mu}_{x} {\mu}_{y}$
Var(X) = sigma_x; Var(Y)=sigma_y

Assume normal distribution and you can show that:

Var(XY)=mu_X^2*sigma_Y+ mu_Y^2*sigma_X+2*mu_Xmu_Y*Cov(X,Y) +sigma_Xsigma_Y+Cov^2(X,Y)

Please Double check, and good luck