Expected Value
Expected Value of Random Variable
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.
$E[X] = \int_{x_{min}}^{x_{max}} x \times f(x) dx$
Where
$f(x)$
is the PDF of
$X$