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.
Where equal to the values that takes and is the probability that takes the value
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.
Where is the PDF of