The expected value of a random variable is the weighted average of all possible values of the variable. The weight here means the probability of the random variable taking a specific value.

Expected Value of Discrete Random Variable

$E[X] = \sum x_i \times P(x_i)$

Where $x_i$equal to the values that $X$takes and $P(x_i)$ is the probability that $X$ takes the value $x_i$

Expected Value of Continuous Random Variable

Since continuous random variables can take uncountably infinitely many values, we cannot talk about a variable taking a specific value. We rather focus on value ranges.

In order to calculate the probability of value ranges, probability density functions (PDF) are used.

PDF is a function that specifies the probability of a random variable taking value within a particular range.

$E[X] = \int_{x_{min}}^{x_{max}} x \times f(x) dx$