Continuous random expectation

If we have a continuous random variable, we can calculate its expectation (essentially, mean) using this formula:

E(X)=\int_a^b x\times f(x)\,\,dx

a and b represent the range of the continuous random variable.

Deriving the formula

If you imagine the distribution graph for a continuous random variable, the area under the graph between a range (say, a<x<b) represents the probability of selecting a value in that range.

To find the expected value then, we can just integrate x multiplied by f(x), the probability density function - because x represents the value and we’re finding the mean.