Probability theory 2

So that these posts don’t get too long, I’ll split them up when there’s more theorems/examples/etc. I want to add. The first new topic is the “law of the unconscious statistician”:

Theorem 1 If {X} is a random variable with {\mathbb E[|X|]<\infty} and {g:\mathbb R\rightarrow\mathbb R} is a Borel-measurable function such that {\mathbb E[|g(X)|]<\infty}, then

\displaystyle \mathbb E[g(X)] = \int_{\mathbb R} g(x)\mathsf dF(x),

where {F} is the distribution function of {X}.

Proof: Define {Y=g(X)}, that is, for each {\omega\in\Omega}, {Y(\omega)=g(X(\omega))}. Then the distribution of {Y} is

\displaystyle  F_Y(y)=\mathbb P(\omega\in\Omega:Y(\omega)\leqslant y) = \mathbb P(\omega\in\Omega:g(X(\omega))\leqslant y) = \mathbb P(X\in A),

where

\displaystyle A=\{x\in\mathbb R: g(x)\leqslant y\}.

Note that {A} is a Borel-measurable set because {g} is a Borel-measurable function. For all such sets {A}, define

\displaystyle \widetilde{\mathbb P}(A) = \int_A \mathsf dF(x).

Then {\widetilde{\mathbb P}} is a probability measure on {\mathcal B(\mathbb R)}, and

\displaystyle \widetilde{\mathbb P}(x\in\mathbb R:g(x)\leqslant y)=F_Y(y).

It follows that

\displaystyle \mathbb E[g(X)] = \int_{\Omega}g(X)\mathsf d\mathbb P = \int_{\mathbb R}g\;\mathsf d\widetilde{\mathbb P} = \int_{\mathbb R}g(x)\mathsf dF(x).

\Box

When {g(x)=x}, we have the familiar formula

\displaystyle \mathbb E[X] = \int_{\mathbb R}x\mathsf dF(x).

When {X} takes countably many values (is discrete), this reduces to

\displaystyle \mathbb E[X]=\sum_{x\in E}x\mathbb P(X=x)

where {E} is the set of values that {X} takes. When {F} is absolutely continuous (i.e. {X} is a continuous random variable that admits a density), this reduces to

\displaystyle \mathbb E[X] = \int_{\mathbb R}xf(x)\mathsf dx,

where {f(x)=\frac{\mathsf d}{\mathsf dx}F(x)} is the density of {X}.

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