Here many people stop learning about probability. This is sort of any annoying place to stop. At this point we are stuck with this muddled idea of a random variable (ie a variable that behaves randomly), that by this level in mathematical progress should seem pretty confusing. The randomly behaving variable is okay for students that don’t have much experience with math, but variables should not be ‘random’. What started as a teaching aid has become something that’s both magical and confusing.
Additionally we have the problem that we need two separate models for Discrete and Continuous Probability Distributions. An even bigger issue is that we never talked about a third type of probability distribution that involves both discrete and continuous components!
Fortunately, we have the answers to all these issues in the development of rigorous probability with measure theory! We introduce the formalized idea of a Random Variable, generalize both discrete and continuous probabilities as a sample space Ω (Omega), and use the Lebesgue Integral to sum up over the sample space. Formally Ω is a set of possible events. And the Lebesgue Integral can be understood simply as a generalization of the Integral covered in basic calculus that is more robust. Our mu is finally E[X] and our generalized form of expectation is:
E[X]=∫ΩX(ω)P(dω)