Expectation Algebra

Recall that expectation is defined as:

$$E(X)={\mu }_X=\sum xp(x)$$

For a constant $a$, we have

$$\begin{array}{rl}& E(a)=a\\ & E(aX)=aE(X)\\ & E(a\pm X)=a\pm E(X)\\ & E(a\pm bX)=a\pm bE(X)\end{array}$$

If we introduce another random variable $Y$:

$$\begin{array}{rl}& E(X\pm Y)=E(X)\pm E(Y)\\ & E(XY)=E(X)E(Y)\end{array}$$

Variance and Covariance Expectation Rules

In general:

$$\begin{array}{rl}V(X)={\sigma }_X^2& =E((X-{\mu }_X{)}^2)\\ & =E(X^2)-{\mu }_X^2\end{array}$$

and

$$\begin{array}{rl}& \\ COV(X,Y)& =E((X-{\mu }_X)(Y-{\mu }_Y))\\ & =E(XY)-{\mu }_X{\mu }_Y\end{array}$$

For some constant $a$:

$$\begin{array}{rl}& V(a)=0\\ & V(a\pm X)=V(X)\\ & V(aX)=a^{2}V(X)\end{array}$$

A constant doesn’t vary, so the variance of a constant is zero. Adding a constant to a variable also doesn’t change its variance.

If we have two random variables and they are independent:

$$COV(X,Y)=0$$

In general, if the variables are not independent:

$$\begin{array}{rl}& V(X\pm Y)=V(X)+V(Y)\pm 2COV(X,Y)\\ & V(XY)\ne V(X)V(Y)\end{array}$$