**Unformatted text preview: **s Inequality) If X ∈ L2 , then
E(X 2 )
P {|X | ≥ a} ≤
a2
for every a > 0. 19–3 Statistics 851 (Fall 2013)
Prof. Michael Kozdron October 23, 2013 Lecture #20: Proofs of the Main Expectation Theorems
Our goal for today is to start proving all of the important results for expectation that were
stated last lecture. Note that the proofs generally follow the so-called standard machine;
that is, we ﬁrst prove the result for simple random variables, then extend it to non-negative
random variables, and ﬁnally extend it to general random variables. The key results for
implementing this strategy are Proposition 18.1 and Proposition 18.2.
Theorem 20.1. Let (Ω, F , P) be a probability space. If L1 denotes the space of integrable
random variables, namely
L1 = {random variables X such that E(X ) < ∞},
then L1 is a vector space and expectation is a linear operator on L1 . Moreover, expectation is
monotone in the sense that if X and Y are random variables with 0 ≤ X ≤ Y and Y ∈ L1 ,
then X ∈ L1 and 0 ≤ E(X ) ≤ E(Y ).
Proof. Suppose that X ≥ 0 and Y ≥ 0 are non-negative random variables,...

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