Expectation

Definition

$$E[X] = \sum_{k} k Prob[X=k]$$

E[X] is a number in the middle range of possible values of the R.V. X. Because in physics the center of gravity of a distribution is called the first moment of inertia of the distribution, E[X] is also called the first moment of the distribution of X.

Rules

• $E(X+Y) = E(X) + E(Y)$ for any two random variables X and Y.

• $E[a X + b] = a E[X] + b$, where a and b are real valued constants.

• But $E[XY] = E[X] E [Y]$ ONLY WHEN X and Y are independent random variables. Therefore, $E[X^2]$ is NOT eqaul to $(E[X])^2$.

Examples

• Let Y be a binomial distribution (see the section ). Calculate $E[Y]$
  
  
  
  
  
  
  
  

• Define R.V. X as the result of flipping a fair coin (tail=0, head=1) and R.V. Y as the result of a single throw of a die. Define a new R. V. $Z = 2 * X + Y$. Calculate $E[X], E[Y], E[Z], E[Z^2]$.