The technical definition of expectation Eg(X) is the sum of P(X=x) g(x) over all possible outcomes x. In the case of a discrete variable, this is an actual sum. In the case of a continuous variable, the sum becomes an integral over the infinitesmal probabilities. We provide the definition and illustrate with examples.
CAUTIONARY NOTE: When the values of the function g(X) are unbounded, the expectation can become + ∞ or - ∞ or the sum of both, which is indeterminate. These cases require special handling, and will mostly be ignored without explicit mention in our treatment below.
S2 Binomial
S3 Poisson
S4 Gamma
S5 Normal