Expected Value

probability-theory

Definition

Expected Value

The expected value is a weighted mean (average) of the possible outcomes of a random variable .

Properties

Linearity

The expectation is linear, i.e. for all random variables and scalars holds:

Independence

If the two random variables and are independent, then:

Law of the Unconscious Statistician

For a measurable function (assuming absolute convergence) holds:

1. Discrete:
2. Continuous:

Discrete vs Continuous

Discrete

The expected value of a discrete random variable with values and probability mass function is given by:

Continuous

The expected value of a continuous random variable with density is given by:

Note that absolutely convergence is assumed. Else, the expectation may be or does not exist.