Expectation of Real-Valued Measurable Function composed with Absolutely Continuous Random Variable
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Theorem
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $X$ be an absolutely continuous random variable on $\struct {\Omega, \Sigma, \Pr}$.
Let $P_X$ be the probability distribution of $X$.
Let $\map \BB \R$ be the Borel $\sigma$-algebra of $\R$.
Let $h : \R \to \R$ be a $\map \BB \R$-measurable function.
Let $f_X$ be a probability density function for $X$.
Let $\lambda$ be the Lebesgue measure on $\struct {\R, \map \BB \R}$.
Then $\map h X$ is integrable if and only if:
- $\ds \int_\R \size {\map h x} \map {f_X} x \rd \map \lambda x < \infty$
and in this case:
- $\ds \expect {\map h X} = \int_\R \map h x \map {f_X} x \rd \map \lambda x$
Proof
From Composition of Measurable Mappings is Measurable:
- $\map h X$ is $\Sigma$-measurable.
So:
- $\map h X$ is a real-valued random variable.
From Characterization of Integrable Functions, we have that:
- $\map h X$ is integrable if and only if $\size {\map h X}$ is integrable.
We have:
\(\ds \int \size {\map h X} \rd \Pr\) | \(=\) | \(\ds \int \size h \circ X \rd \Pr\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int_\R \size {\map h x} \rd \map {P_X} x\) | Integral with respect to Pushforward Measure, Definition of Probability Distribution | |||||||||||
\(\ds \) | \(=\) | \(\ds \int_\R \size {\map h x} \map {f_X} x \rd \map \lambda x\) | Change of Measures Formula for Integrals, Definition of Probability Density Function |
Then:
- $\ds \int \size {\map h X} \rd \Pr$ if and only if $\ds \int_\R \size {\map h x} \map {f_X} x \rd \map \lambda x < \infty$
so:
- $\size {\map h X}$ is integrable if and only if $\ds \int_\R \size {\map h x} \map {f_X} x \rd \map \lambda x < \infty$.
So:
- $\map h X$ is integrable if and only if $\ds \int_\R \size {\map h x} \map {f_X} x \rd \map \lambda x < \infty$.
If $\map h X$ is integrable, then we have:
\(\ds \expect {\map h X}\) | \(=\) | \(\ds \int \map h X \rd \Pr\) | Definition of Expectation | |||||||||||
\(\ds \) | \(=\) | \(\ds \int_\R \map h x \rd \map {P_X} x\) | Integral with respect to Pushforward Measure, Definition of Probability Distribution | |||||||||||
\(\ds \) | \(=\) | \(\ds \int_\R \map h x \map {f_X} x \rd \map \lambda x\) | Change of Measures Formula for Integrals: Corollary |
$\blacksquare$
Sources
- 1991: David Williams: Probability with Martingales ... (previous) ... (next): $6.12$: The 'elementary formula' for expectation