## Expectation Value

In probability and statistics, the *expectation* or *expected value*, is the weighted average value of a random variable.

### Expectation of continuous random variable

*E*(*X*) is the expectation value of the continuous random variable *X*

*x* is the value of the continuous random variable *X*

*P*(*x*) is the probability density function

### Expectation of discrete random variable

*E*(*X*) is the expectation value of the continuous random variable *X*

*x* is the value of the continuous random variable *X*

*P*(*x*) is the probability mass function of *X*

### Properties of expectation

##### Linearity

When a is constant and X,Y are random variables:

*E*(*aX*) = *aE*(*X*)

*E*(*X*+*Y*) = *E*(*X*) +* E*(*Y*)

##### Constant

When c is constant:

*E*(*c*) = *c*

##### Product

When X and Y are independent random variables:

*E*(*X ⋅Y*) = *E*(*X*)* ⋅ E*(*Y*)

### conditional expectation