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

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