Included in at least one reviewed higher-education syllabus.
S4-5.2 Operations on Random Variables
You can compute moments, apply Chebyshev and Markov inequalities, use characteristic and moment-generating functions, and work with jointly Gaussian variables.
Expectation compresses a whole distribution into a few numbers — mean, variance and their higher-order relatives — and the Chebyshev and Markov inequalities say how much those numbers guarantee even when the distribution itself is unknown. Characteristic and moment-generating functions look like abstraction for its own sake until they crack sums of random variables without effort, which is exactly the road to the Central Limit Theorem. Jointly Gaussian variables close the unit as the one multivariate case where correlation tells the entire story. The confusion to avoid is treating moments as the distribution; two very different distributions can share a mean and variance.
Work through these
Expectation; moments about origin and central moments
Expectation compresses a whole distribution into a few numbers, and moments are the family those numbers belong to: the mean locates it, the variance spreads it, and higher moments describe its shape. Almost every practical result is stated in these terms.
Chebyshev's and Markov's inequalities
These inequalities bound how much probability can sit far from the mean using only the mean and variance, without knowing the distribution at all. That is what makes them useful when the distribution is unknown or awkward.
Characteristic function and moment-generating function
These transforms look like abstraction for its own sake until they crack sums of random variables with almost no work. That is exactly the route by which the Central Limit Theorem is reached.
Central Limit Theorem (statement and meaning)
Add enough independent random variables and the total tends toward a Gaussian shape regardless of what you started with. This is why the Gaussian appears everywhere in engineering, and knowing what the theorem does not promise matters as much as what it does.
Expected value of functions of multiple random variables; joint moments
Extending expectation to functions of several random variables, and the joint moments that result, of which covariance is the one you will use most. Correlation between signals is defined from exactly here.
MIT OCW 18.440: Probability and Random Variables · CourseJointly Gaussian random variables and their properties
Jointly Gaussian variables are the one multivariate case where correlation tells you everything about the dependence between them. That property is why so much of communications and estimation assumes Gaussian and stops there.
MIT OCW 18.440: Probability and Random Variables · Course
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