# Expectation A random variable Informally, it can be considered the mean of the possible values that a random variable can take. > [!summary] **Definition:** Expectation > The **expectation** of a random varaible $X$, written $E[X]$, is the *average of all the values* the random variable can take on, each *weighted by the probability* that the random variable will take on that value. > $\mathbb{E}[X]=\sum_{x} x \cdot P(X=x)$ Expectation goes by many other names: mean, weighted average, center of mass, 1st moment, etc. ## Properties of Expectations