Probability theory and stochastic processes Theorem

Law of the unconscious statistician

Builds on Expected value Law of a random variable Probability mass function Probability density function

Statement

Let $X$ be a random variable on $(\Omega, \mathcal{F}, P)$ with values in $(E, \mathcal{E})$ and with law $\mu_X$ . Let $g \colon E \to \mathbb{R}$ be measurable. Then $g(X)$ is integrable if and only if $\int_E |g| \, d\mu_X < \infty$ . In that case, $E[g(X)] = \int_E g(x) \, \mu_X(dx).$ In particular, if $X$ is discrete with mass function $p$ , then $E[g(X)] = \sum_x g(x) \, p(x)$ . If $X$ is real with density $f$ , then $E[g(X)] = \int_{-\infty}^{\infty} g(x) f(x) \, dx$ .

Remarks

The formula also holds for $g \ge 0$ without any integrability condition, with both sides in $[0, \infty]$ . The proof checks the formula first for indicator functions $g = 1_B$ , where it says $P(X \in B) = \mu_X(B)$ . Then it extends to simple, nonnegative and integrable $g$ . The point is that one does not need the law of $g(X)$ to compute $E[g(X)]$ .

History

The name refers to statisticians who use the formula as the definition of $E[g(X)]$ . They are "unconscious" that it is a theorem that needs a proof.

Reading path

To understand this page from the ground up, read these 49 pages in order. The foundations come first.

Construction of the reals from Cauchy sequences concept
Element definition
Integers object
Ordered field definition
Proposition concept
Random experiment concept
Rational numbers object
Real numbers object
Set definition
Ring of integers statement
Disjunction definition
Implication definition
Predicate definition
Ordered pair definition
Union definition
Sample space definition
The rationals form a field statement
Existential quantifier definition
Universal quantifier definition
Cartesian product definition
Subset definition
The rationals form an ordered field statement
Uniqueness quantifier definition
Binary relation definition
Universal set definition
Order on the reals definition
Power set definition
Complement definition
Function definition
Interval definition
Injective function definition
Preimage of a set definition
Sigma-algebra definition
Countable set definition
Event definition
Borel sigma-algebra definition
Probability axioms concept
Probability measure definition
Probability distribution definition
Probability space definition
Random variable definition
Cumulative distribution function definition
Discrete random variable definition
Law of a random variable definition
Random variable with a density definition
Probability mass function definition
Expected value definition
Probability density function definition
Law of the unconscious statistician statement

Where this leads

33 pages build on this one.

Tag	`01ZWX875AT`
URI	`m2://01ZWX875AT`
Aliases	`lotus`, `change-of-variables-formula-for-expectation`
Kind	statement, theorem
Writing	W0 stub
MSC	60A05
Status	This copy of the site has no build data. Run `m2 status 01ZWX875AT`.
Uses	Expected value, Law of a random variable, Probability mass function, Probability density function
Same as	Wikidata Q6503509 Wikipedia Law of the unconscious statistician

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