20.2 Continuous random variables

So far, we have only defined the expected value for discrete random variables. As 𝔼[X] describes the average value of X in the long run, it should exist for continuous random variables as well.

The interpretation of the expected value was simple: outcome times probability, summed over all potential values. However, there is a snag with continuous random variables: we don’t have such a mass distribution, as the probabilities of individual outcomes are zero: P(X = x) = 0. Moreover, we can’t sum uncountably many values.

What can we do?

Wishful thinking. This is one of the most powerful techniques in mathematics, and I am not joking.

Here’s the plan. We’ll pretend that the expected value of a continuous random variable is well-defined, and let our imagination run free. Say goodbye to mathematical precision, and allow our intuition to unfold. Instead of the probability of a given outcome, we can talk about X landing in a...