851_lectures17_24

# Therefore using theorem 203 we conclude ey ex 12 213

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: r the expectations of discrete and continuous random variables encountered in introductory probability follow from the general theory developed. The ﬁrst examples we will discuss, however, are the calculations of expectations directly from the deﬁnition and theory. Example 21.3. Consider ([0, 1], B1 , P) where B1 are the Borel sets of [0, 1] and P is the uniform probability. Let X : Ω → R be given by X (ω ) = ω so that X is a uniform random variable. We will now compute E(X ) directly by deﬁnition. Note that X is not simple since the range of X , namely [0, 1], is uncountable. However, X is positive. This means that as a consequence of Propositions 18.1 and 18.2, if Xn is a sequence of positive, simple random variables with Xn ↑ X , then E(Xn ) ↑ E(X ). Thus, let ￿ 0, 0 ≤ ω < 1/2, X1 (ω ) = 1/2, 1/2 ≤ ω ≤ 1 0 ≤ ω ≤ 1/4, 0, 1/4, 1/4 ≤ ω < 1/2, X2 (ω ) = 1/2, 1/2 ≤ ω < 3/4, 3/4, 3/4 ≤ ω ≤ 1, 21–2 and in general, Xn (ω ) = 2n − 2 ￿ j =1 j−1 1 2n ￿ j−1 j ≤ω< n n 2 2 ￿ 2n − 1 + 1 2n ￿ 2n − 2 2n − 1 ≤ω≤ 2n 2n ￿ . Thus...
View Full Document

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

Ask a homework question - tutors are online