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A probability space is a collection (Ω, F, P) with the following structure:
- (i) Ω is a set, which we call the sample space.
- (ii) F is a collection of subsets of Ω. An element of F is called an event.
- (iii) The probability measure P is a function from F to [0, 1]. It assigns a probability
- to each event in F
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The set F of events should satisfy certain natural conditions:
- (1) Ω ∈ F.
- (2) If F contains a set A, then it also contains the complement Ac (i.e. Ω A).
- (3) If (Ai, i ∈ I) is a finite or countably infinite collection of events in F, then their union U i∈I Ai is also in F
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the probability measure P should satisfy the following conditions (the probability
axioms):
- (1) P(Ω) = 1
- (2) If (Ai, i ∈ I) is a finite or countably infinite collection of disjoint events, then
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Any distribution function F must obey the following properties:
- (1) F is non-decreasing.
- (2) F is right-continuous.
- (3) F(x) → 0 as x → −∞.
- (4) F(x) → 1 as x → ∞
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A random variable X is discrete if
there is a finite or countably infinite set B such that P(X ∈ B) = 1
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A random variable X is called continuous if
its distribution function F can be written as an integral. That is, there is a function f such that
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basic properties of expectation:
- (1) For any event A, write IA for the indicator function of A. Then E IA = P(A).
- (2) If P(X ≥ 0) = 1, then E X ≥ 0.
- (3) (Linearity 1): E (aX) = aE X for any constant a.
- (4) (Linearity 2): E (X + Y ) = E X + E Y .
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