Variance and Covariance of Linear Transformations

Question

Suppose $X$ and $Y$ are random variables with $E(X)=2, E(Y)=3 Var(X)= 4, Var(Y)=10$ and $Cov(X,Y)=-5$

1. Find $Var (5X+2Y)$

From my book I know $$Var(5X+2Y)= Var(5X)+Var(2Y)+2Cov(5X,2Y)$$ but after that do I just substitute in values so $Var(5X)= 5\cdot 4=20$? If so, how do I figure out $Cov(5x,2y)$?

2. Find $Cov(3X+Y,Y)$

Answer 1

Hints Note that $$Var (kZ) = \mathbb{E}[Z^2] - \mathbb{E}[Z]^2 = \mathbb{E}[(kZ)^2] - \mathbb{E}[kZ]^2 = \mathbb{E}[k^2 Z^2] - (k\mathbb{E}[Z])^2$$ Can you finish this and express $Var(kZ)$ in terms of $k$ and $Var Z$?

Then do the same thing to the $Covar(X,Y)$, directly using the definition.

Answer 2

Let's compute:

$$Var(5X+2Y)= Var(5X)+Var(2Y)+2Cov(5X,2Y) = 25 Var (X) + 4 Var (Y) + 20 Cov(X,Y) = 25 (4) + 4(10) + 20 (-5) = 40$$

To understand the manipulations, you need to see the properties of the expectation and the definition of covariance.

\begin{align*}\Bbb{E}[a X + bY] &= a \Bbb{E}[X] + b \Bbb{E}[Y]\\ Cov(X,Y) &= \Bbb{E}[(X - \Bbb{E}[X])(Y - \Bbb{E}[Y])]\\ Var(X) &= Cov (X,X) \end{align*}

take a look at https://en.wikipedia.org/wiki/Expected_value,

and https://en.wikipedia.org/wiki/Covariance