Expectation and Variance

The expectation of a random variable X is defined by:

$$E[X] = \left\{ \begin{array}{ll} \int_{-\infty}^{\infty} x f(... ...} x p(x) & \mbox{if $X$\space is discrete} \end{array} \right.$$ (1)

The expectation of a function of X can be calculated as:

$$E[g(X)] = \left\{ \begin{array}{ll} \int_{-\infty}^{\infty} g... ...(x) p(x) & \mbox{if $X$\space is discrete} \end{array} \right.$$ (2)

Some properties of expectations:

E[X+Y] = E[X] + E[Y] (3)

$$E[XY] = E[X]E[Y] \mbox{if $X$\space and $Y$\space are independent}$$ (4)

The variance of a random variable X is:

$$Var[X] = \left\{ \begin{array}{ll} \int_{-\infty}^{\infty} (x... ...{2} p(x) & \mbox{if $X$\space is discrete} \end{array} \right.$$ (5)

If X and Y are independent, then:

Var[X+Y] = Var[X]+Var[Y]. (6)

If X and Y are not independent, then:

Var[X+Y] = Var[X]+Var[Y]+2Cov(X,Y), (7)

where

Cov(X,Y) = E[XY] - E[X]E[Y] (8)