Random Variables

Linear Combinations of RV

Let $X,Y$ be random variables, $a,b,c$ be constants.

Expectation have the following properties for linear transformations:

$E(aX+c)=aE(X)+c$

For a linear combination, we have

$E(aX+bY)=aE(X)+bE(Y)$

Variance is a bit more complicated. Recall that the calculation of variance involves the average difference from the mean squared. It is no surprise that any constant coefficient is squared, and any translations does not impact the spread of data.

$Var(aX+c)=a^{2}Var(X)$

For a linear combination, when the two events are independent,

$Var(aX+bY)=a^{2}Var(X)+b^{2}Var(Y)$

When the events are dependent,

$Var(aX+bY)=a^{2}Var(X)+abCov(X,Y)+b^{2}Var(Y)$