Questions about the expected value of a random variable.

The average value of a randomly chosen quantity is its *expectation* or *expected value*. For example, the expected value of the number you get when you roll a fair 6-sided dice is 3.5.

In general, if $X$ is a random variable defined on a probability $(\Omega, \Sigma, P)$, then the expected value of $X$, denoted by $E[X], \langle X \rangle,$ or $\bar{X}$ is defined as the Lebegue integral

$$E[X]= \int_{\Omega} X(\omega) dP(\omega)$$

The expected value is often the first and most important thing you want to know about a random variable. For example, in a betting game, the best strategy is often the one that maximizes the expected value of the amount you win.

This tag is for questions about:

- Computing the expected value in a specific situation.
- Understanding the properties of expected values, such as Markov's inequality or linearity of expectation.
- Proving theorems about the expected value of abstract random variables.
- Understanding what the expected value means and what it tells you about a random variable.