Expectation and Integration

Expectation is the Lebesgue integral of a random variable against the probability measure, a single notion that unifies sums for discrete variables and integrals for continuous ones and inherits powerful convergence theorems from measure theory.

Definition

The expectation of a random variable is its integral against the probability measure, built first for non-negative variables as a supremum over simple approximations and then extended to integrable variables as the difference of positive and negative parts.

Scope

The topic covers the construction of expectation for simple, non-negative, and integrable random variables, the monotone and dominated convergence theorems and Fatou's lemma, the change-of-variables formula relating expectation to integrals against the distribution, moments and the Lp spaces, and the Jensen, Holder, Markov, and Chebyshev inequalities.

Core questions

• How is expectation defined for an arbitrary random variable, not just discrete or continuous ones?
• Under what conditions can a limit be moved inside an expectation?
• How do moments and the Lp spaces quantify the size of a random variable?
• Which inequalities bound probabilities and expectations in terms of moments?

Key concepts

• expectation as Lebesgue integral
• monotone and dominated convergence
• Fatou's lemma
• moments and variance
• Lp spaces of random variables

Key theories

Monotone and dominated convergence theorems

For increasing non-negative random variables the expectation of the limit equals the limit of expectations, and for sequences dominated by an integrable variable the same interchange holds, giving the limit theorems that the elementary theory lacks.

Jensen's inequality

For a convex function the expectation of the function of a random variable is at least the function of its expectation, which yields moment comparisons, the contraction property of conditional expectation, and many bounds throughout probability.

Markov and Chebyshev inequalities

The probability that a non-negative random variable exceeds a level is bounded by its mean divided by that level, and applied to squared deviations this controls dispersion in terms of variance, providing the elementary route to the weak law of large numbers.

Clinical relevance

Expectation and its inequalities are used everywhere quantities are averaged under uncertainty: they define means, variances, and risk measures in statistics and finance, supply the concentration bounds behind learning theory and randomized algorithms, and provide the convergence theorems that justify Monte Carlo estimation.

History

Once Lebesgue's integral became available, probabilists identified expectation with integration against the probability measure, an identification made explicit in Kolmogorov's framework and developed with its convergence theorems and classical inequalities in the standard graduate texts.

Key figures

• Henri Lebesgue
• Johan Jensen
• Pafnuty Chebyshev
• Andrey Markov

Seminal works

• billingsley1995

Frequently asked questions

Is expectation the same as the average over outcomes?

Yes in spirit: it is the integral of the random variable weighted by the probability of each outcome, which reduces to a weighted sum for discrete variables and to an ordinary integral against a density for continuous ones.

When can I swap a limit and an expectation?

The monotone convergence theorem allows it for increasing non-negative sequences and the dominated convergence theorem allows it when the sequence is bounded by a fixed integrable variable; without such conditions the interchange can fail.