ST 2133 Distribution Theory

    Cards (209)

    • Sample space (Ω)
      The set of all possible outcomes of an experiment
    • Event (A)

      A set of outcomes which is of interest
    • Classical probability

      P(A) = n(A) / n(Ω), where n(A) is the number of outcomes in A and n(Ω) is the total number of equally likely possible outcomes
    • Event A = {sum of two dice is greater than 10}

      • (5, 6)
      • (6, 5)
      • (6, 6)
    • n(Ω)
      Finite sample space
    • The concept of 'classical probability' was first introduced in ST104b Statistics 2
    • Probability is a mathematical construct
    • Whenever we apply the ideas of probability to real-world situations we always make assumptions
    • Probability statements are statements about a mathematical model, not statements about reality
    • Elementary outcomes

      Outcomes of the form (value on first die, value on second die)
    • Event A = {(5, 6), (6, 5), (6, 6)}
    • A ⊂ Ω
    • Assuming fair dice, all members of the sample space are equally likely
    • n(A) = 3, n(Ω) = 36
    • P(A) = 3/36 = 1/12
    • The probability calculated is a statement about a model for rolling two dice in which each of the possible outcomes is equally likely
    • For the model to be valid, the dice must be fair such that the six outcomes from rolling each die are equally likely
    • Slight differences in the likelihood of outcomes may be ignored for simplicity
    • Tossing a fair coin repeatedly until both a head and a tail have appeared at least once

      • ΩH = {i tails followed by 1 head : i = 1, 2, ...}
      • ΩT = {i heads followed by 1 tail : i = 1, 2, ...}
    • P(TTH) + P(HHT) = 2 × 1/2 × 1/3 = 1/4
    • Probability that a child has blue eyes is 0.25
    • Assume independence between children in a family with three children
    • P({A1, A2, A3} : at least two of Ai are 1 | {A1, A2, A3} : at least one of Ai is 1) = 0.2703
    • P({A1, A2, 1} : at least one of A1 and A2 are 1 | A3 = 1) = 0.4375
    • σ-algebra
      A collection of subsets of a set Ψ that satisfies: 1) ∅ ∈ B, 2) if A ∈ B then Ac ∈ B, 3) if A1, A2, ... ∈ B then ∪∞i=1 Ai ∈ B
    • Measurable space
      The pair (Ψ, B) where Ψ is a set and B is a σ-algebra defined on Ψ
    • Measure
      A function m: B → R+ such that: 1) m(A) ≥ 0 for all A ∈ B, 2) m(∅) = 0, 3) if A1, A2, ... ∈ B are mutually exclusive then m(∪∞i=1 Ai) = ∑∞i=1 m(Ai)
    • Measure space
      The triple (Ψ, B, m) where (Ψ, B) is a measurable space and m is a measure on (Ψ, B)
    • Indicator function
      IΩ(x) = 1 if x ∈ Ω, 0 otherwise
    • Counts and indicator functions are examples of measures
    • Probability measure
      A measure P: F → [0, 1] on a measurable space (Ω, F) such that P(Ω) = 1
    • Probability space
      The triple (Ω, F, P) where (Ω, F) is a measurable space and P is a probability measure on (Ω, F)
    • P(A) is the probability that the outcome of the experiment is in A
    • P(Ac) is the probability that the outcome of the experiment is not in A
    • P(A ∩ B) is the probability that the outcome of the experiment is in both A and B
    • P(A ∪ B) is the probability that the outcome of the experiment is in A or B or in both
    • Probability space
      A collection of events (forming a σ-algebra on the sample space) and a probability measure
    • Probability measures

      • P(A) = n(A)/n(Ω)
      • P(A) = IA(ω) for some ω ∈ Ω
    • Outcomes

      Elements of the sample space
    • Events

      Subsets of the sample space