Expectation, Conditioning, and Limit Theorems

Manual: General · Subject: Statistics

Master the machinery of conditional expectation, dependence, and asymptotic convergence.

Expectation as an operator

Linear and conditional expectation

Expectation is an integral operator: $\mathbb{E}[X]=\int x\,dP$ . Conditional expectation $\mathbb{E}[X\mid \mathcal{G}]$ is the best mean-square predictor of $X$ given information $\mathcal{G}$ .

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Useful identity

If $Y$ is $\mathcal{G}$ -measurable, then $\mathbb{E}[YX\mid \mathcal{G}] = Y\,\mathbb{E}[X\mid \mathcal{G}]$ whenever the product is integrable.

Important conditional expectation properties

Tower property

\mathbb{E}[\mathbb{E}[X\mid\mathcal{G}]]=\mathbb{E}[X]

Law of total expectation

\mathbb{E}[X]=\mathbb{E}[\mathbb{E}[X\mid Y]]

Law of total variance

\mathrm{Var}(X)=\mathbb{E}[\mathrm{Var}(X\mid Y)]+\mathrm{Var}(\mathbb{E}[X\mid Y])

What does the tower property express?

Conditional variance decomposition Nested conditional expectations collapse correctly Independence of all random variables Symmetry of joint distributions

State the weak law of large numbers in one sentence.

Asymptotic modes of convergence

Common asymptotic notions include almost sure convergence, convergence in probability, convergence in distribution, and convergence in $L^p$ . Each has different strength and use in proving consistency and limit laws.

Which convergence is strongest among the following?

In distribution In probability Almost surely They are all equivalent

What is the central limit theorem's basic conclusion for i.i.d. data with finite variance?

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Expectation, Conditioning, and Limit Theorems

Expectation as an operator

Linear and conditional expectation

Asymptotic modes of convergence