Recap Part A Prob

  1. 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
  2. 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
  3. 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, thenImage Upload 1
  4. 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 → ∞
  5. A random variable X is discrete if
    there is a finite or countably infinite set B such that P(X ∈ B) = 1
  6. 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 thatImage Upload 2
  7. 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 .
Author
Nat1234
ID
334919
Card Set
Recap Part A Prob
Description
part A probability chapter 1
Updated