Axioms of expectation:(adsbygoogle = window.adsbygoogle || []).push({});

1. X>0 => E(X)>0

2. E(1)=1

3. E(aX+bY) = aE(X) + bE(Y)

4. If X_n is a nondecreasing sequence of numbers and lim(X_n) = X, then E(X)=lim E(X_n) [montone convergence theorem]

Definition: P(A)=E[I(A)]

Using the above, prove that if X=0 almost surely [i.e. P(X=0)=1 ], then E(X)=0.

Proof:

X=0 almost surely <=> |X|=0 almost surely

[note: I is the indicator/dummy variable

I(A)=1 if event A occurs

I(A)=0 otherwise]

|X| = |X| I(|X|=0) + |X| I(|X|>0)

=> E(|X|) = E(0) + E[|X| I(|X|>0)]

=E(0*1) + E[|X| I(|X|>0)]

=0E(1) + E[|X| I(|X|>0)] (axiom 3)

=0 + E[|X| I(|X|>0)] (axiom 2)

=E[|X| I(|X|>0)]

=E[lim |X| * I(0<|X|<N)] (lim here means the limit as N->∞)

=lim E[|X| * I(0<|X|<N)] (axiom 4)

=lim E[N * I(0<|X|<N)]

=lim N * E[I(0<|X|<N)]

=lim N * P(0<|X|<N) (by definition)

=lim (0) since P(X=0)=1 => P(0<|X|<N)=0

=0

=>E(X)=0

=======================================

Now, I don't understand the parts in red.

1. We have |X| I(|X|=0), taking the expected value it becomes E(0), how come?? I don't understand why E[|X| I(|X|=0)] = E(0).

2. lim E[|X| * I(0<|X|<N)] = lim E[N * I(0<|X|<N)], why??

Thanks for explaining!

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# X=0 almost surely => E(X)=0

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