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**Unformatted text preview: **j 1Aj . j =1 Observe that if pj = 0 for inﬁnitely many j , then X is simple and can be written as
X (ω ) = j
1Aj j =1 for some n < ∞ which implies that
E(X ) = j
j =1 P {Aj } = n
j =1 j P {X = j } . On the other hand, suppose that pj = 0 for inﬁnitely many j . In this case, X is not simple.
We can approximate X by simple functions as follows. Let
Aj,n = {ω : X (ω ) = j, j ≤ n}
so that Aj,n ⊆ Aj,n+1 and ∞
Aj,n = Aj . n=1 22–1 That is, Aj,n ↑ Aj and so by continuity of probability we conclude
lim P {Aj,n } = P {Aj } . n→∞ If we now set
Xn (ω ) = n
X (ω )1Aj,n = j =1 so that E(Xn ) = n
j 1Aj,n j =1 n
j =1 j P {Aj,n } , then (Xn ) is a sequence of positive simple random variable with Xn ↑ X . Proposition 18.2
then implies
E(X ) = lim E(Xn ) = lim
n→∞ n→∞ n
j =1 j P {Aj,n } = ∞
j =1 P {Aj } = ∞
j =1 j P {X = j } as required. Computing Expectations of Continuous Random Variables
We will now turn to that other formula for computing expectatio...

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