851_lectures17_24

# 851_lectures17_24

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: that X and Y are general random variables and let α ∈ R. Since ( α X ) = (α X ) + − ( α X ) − ≤ ( α X ) + + ( α X ) − ≤ | α | ( X + + X ) and (X + Y ) = ( X + Y )+ − (X + Y )− ≤ (X + Y )+ + ( X + Y )− ≤ X + + X − + Y + + Y − . Hence, since X + , X − , Y + , Y − ∈ L1 , we conclude that αX ∈ L1 and X + Y ∈ L1 . Finally, since E(X ) = E(X + ) − E(X − ) by deﬁnition, and since we showed above that expectation is linear on non-negative random variables, we conclude that expectation is linear on general random variables. 20–1 Theorem 20.2. If X : (Ω, F , P) → (R, B ) is a random variable, then X ∈ L1 if and only if |X | ∈ L 1 . Proof. Suppose that X ∈ L1 so that E(X + ) < ∞ and E(X − ) < ∞. Since |X | = X + + X − and since expectation is linear, we conclude that E(|X |) = E(X + + X − ) = E(X + ) + E(X − ) < ∞ so that |X | ∈ L1 . On the other hand, suppose that |X | ∈ L1 so that E(X + ) + E(X − ) < ∞. However, since X + ≥ 0 and X −...
View Full Document

## This document was uploaded on 04/07/2014.

Ask a homework question - tutors are online