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**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...

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