Properties of Expectation 11/6

In general for any function of the random variable we have the formula:

$$E[g(X)]=\sum_{i}g(x_i) \times m(x_i) \qquad .$$

$$E[aX+b]=aE[X]+b,\mbox{ where a and b are real valued constants}$$

Beware:
If we need to take a function of a random variable, we cannot in general say that the expectation of the function is the function of the expectation, look for instance at the function g(X)=X², for the random variable X which takes on -1 with probability 1/5, +1 with probability .3, and 0 with probability 1/2.

For this random variable E[X]=.1 and E[X²]=.5 which is NOT the same as .01=(E[X])².

  We define the kth moment as

$$E[X^k]=\sum x^k p(x)$$

For any two random variables X and Y, we have:

$$E(X+Y)=E(X)+E(Y) \qquad$$

Only for independent random variables do we have

$$If\;\; X\; and \; Y\; independent\; E(XY)=E(X)E(Y)$$