Covariance Proof for X,Y: aX + b, Y+d & aX+bY, cX+dY

[TomJerry]

If X and Y are two random variable , then the covariance between them is defined as Cov(X,Y) = E[XY] - E(X)E(Y)


i) Show that $Cov (aX + b , (Y + d)) = ac Cov(X,Y)$

ii) $Cov(aX + bY, cX + dY) = ac \sigma_x ^2 + bd \sigma_y ^2 +(ad + bc) Cov(X,Y)$

 

[statdad]

Use properties of expectations to expand the portions. For example, for your ``i'':

$$\begin{align*} cov(aX+b, cY+d) & = E[(aX+b)(cY+d)] - E[(aX+b)]E[(cY+d)] \\ & = E[acXY + adX + bcY + bd] - (a\mu_X+b)(b\mu_Y+d) \end{align*}$$

Continue on from there - you know what the answer should look like when you have combined all terms and simplified. A similar approach works for ``ii''.