# Recap Part A Prob

 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 probabilityto each event in F 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 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 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 → ∞ A random variable X is discrete if there is a finite or countably infinite set B such that P(X ∈ B) = 1 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 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 . AuthorNat1234 ID334919 Card SetRecap Part A Prob Descriptionpart A probability chapter 1 Updated2017-10-09T14:47:42Z Show Answers