851_lectures17_24

# 8 proposition 172 if x and y are simple random

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Unformatted text preview: C4 = ( 1 , 3 ) , 24 17–1 C5 = [ 3 , 7 ) , 48 C6 = { 7 }, 8 C7 = ( 7 , 1]. 8 Proposition 17.2. If X and Y are simple random variables, then E(α X + β Y ) = α E(X ) + β E(Y ) for every α, β ∈ R. Proof. Suppose that X and Y are simple random variables with X= n ￿ ai 1Ai and Y = i=1 m ￿ bj 1 Bj j =1 where A1 , . . . , An ∈ F and B1 , . . . , Bm ∈ F each partition Ω. Since αX = α n ￿ ai 1 A i = i=1 n ￿ (αai )1Ai i=1 we conclude by deﬁnition that E(α X ) = n ￿ i=1 (αai )P {Ai } = α n ￿ i=1 ai P {Ai } = αE(X ). The proof of the theorem will be completed by showing E(X + Y ) = E(X ) + E(Y ). Notice that {Ai ∩ Bj : 1 ≤ i ≤ n, 1 ≤ j ≤ m} consists of pairwise disjoint events whose union is Ω and X +Y = n m ￿￿ i=1 j =1 (ai + bj )1Ai ∩Bj . Therefore, by deﬁnition, E(X + Y ) = = = n m ￿￿ i=1 j =1 n m ￿￿ i=1 j =1 n ￿ i=1 (ai + bj )P {Ai ∩ Bj } ai P {Ai ∩ Bj } + ai P {Ai } + m ￿ j =1 n m ￿￿ i=1 j =1 bj P {Ai ∩ Bj } b j P {B j } and the proof is complete. Fact. If X and Y are simple random variables with X ≤...
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