Mathematical Expectation of Discrete Probability Distribution

The mean, as a measure of central tendency, gives a rough sense of the middle or average value taken by a random variable across an experiment. For a random variable, this idea is formalised as mathematical expectation, also called the expected value.

Suppose a discrete random variable $X$ can take finite values $x_1, x_2, x_3, \dots, x_n$ with respective probabilities $p_1, p_2, p_3, \dots, p_n$, where the probabilities sum to $1$. The mathematical expectation of $X$ is the weighted average of its possible values, each value weighted by the probability of it occurring.