ST259 Probability Test I Review
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ST259 Probability Test I Review

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Questions and Answers

What is the formula used to compute the probability of the union of three events E1, E2, and E3?

  • P(E1 | E2) + P(E2 | E3)
  • P(E1 ∩ E2 ∩ E3)
  • P(E1) + P(E2) + P(E3)
  • P(E1) + P(E2) + P(E3) - P(E1 ∩ E2) - P(E1 ∩ E3) - P(E2 ∩ E3) (correct)

    • How does the multiplication rule apply to independent events?

  • P(A ∩ B) = P(A) + P(B)
  • P(A ∩ B) = P(A) * P(B) (correct)
  • P(A ∩ B) = P(A) / P(B)
  • P(A ∩ B) = P(A | B) + P(B | A)

    • What does the probability mass function (pmf) of a discrete random variable provide?

  • The cumulative probabilities of outcomes
  • A continuous distribution of outcomes
  • A method for deriving variances directly
  • The probabilities of each outcome being observed (correct)

    • How can you derive the pmf from the cumulative distribution function (cdf)?

    <p>By calculating the difference between the cdf values at successive points</p> Signup and view all the answers

    Which of the following represents the expected value of a discrete random variable X multiplied by a constant a and added to a constant b?

    <p>E(aX + b) = aE(X) + b</p> Signup and view all the answers

    What is the complement rule according to the probability function?

    <p>P(A^c) = 1 - P(A)</p> Signup and view all the answers

    Which formula represents the addition rule for two events A and B?

    <p>P(A ∪ B) = P(A) + P(B) - P(A B)</p> Signup and view all the answers

    What does the domination principle state regarding probabilities?

    <p>A ⊂ B implies P(A) ≤ P(B)</p> Signup and view all the answers

    What is the law of total probability used for?

    <p>To find the unconditional probability of an event based on partitioning the sample space.</p> Signup and view all the answers

    Which statement correctly describes independent events A and B?

    <p>P(A | B) = P(A)</p> Signup and view all the answers

    What is the inclusion-exclusion rule for three events A1, A2, and A3?

    <p>P(A1 ∪ A2 ∪ A3) = P(A1) + P(A2) + P(A3) - P(A1 ∩ A2) - P(A1 ∩ A3) - P(A2 ∩ A3)</p> Signup and view all the answers

    What is the difference rule for probability functions?

    <p>P(A - B) = P(A) - P(A ∩ B)</p> Signup and view all the answers

    When proving Bayes' Formula, which principle is fundamentally employed?

    <p>The law of total probability</p> Signup and view all the answers

    Study Notes

    ST259 Test I Information

    • Date: October 11, 2024
    • Time: 4:00 p.m.
    • Location: LH3094
    • Duration: 75 minutes
    • Aids Allowed: Non-programming calculators and a formula sheet provided on the test paper.
    • Content: Material from Parts 1, 2, and 3 of Lecture Notes
    • Test Structure:
      • Theoretical Questions (20%-25%):
        • Properties of the probability function:
          • Complement rule
          • Domination principle
          • Addition rule
          • Inclusion-exclusion rule
          • Difference rule
        • Properties of the conditional probability function:
          • Complement rule
          • Domination principle
        • General multiplication rule
        • Law of total probability and Bayes' Formula
        • Independence of events
        • Fundamental probability formula for discrete random variables
      • Practical Questions (75%-80%):
        • Sample space and events
        • Classical probability
        • Conditional probability
        • Independence of events
        • PMF and CDF of discrete random variables
        • Mathematical expectation of discrete random variables
    • Additional Notes:
      • Review lecture notes, WeBWorK and lab assignments, and recommended homework problems for a comprehensive set of examples.

    Useful Formulae

    • Inclusion-Exclusion Rule for Three Events:

      • P(A1 ∪ A2 ∪ A3) = P(A1) + P(A2) + P(A3) − P(A1 A2) − P(A1 A3) − P(A2 A3) + P(A1 A2 A3)

    • Multiplication Rule for Independent Events:

      • P(A1 ∩ A2 ∩ · · · ∩ An) = P(A1) · P(A2) · P(A3) · · · P(An)

    • Bayes’ Formula:

      • P(Bk | A) = P(A | Bk)P(Bk) / (P(A | B1)P(B1) + · · · + P(A | Bn)P(Bn))

    • Binomial Coefficient:

      • nCk,n = n! / (k!(n − k)!)

    • Fundamental Probability Formula for Discrete Random Variables:

      • P(X ∈ A) = Σx∈A∩D p(x)

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    ST259 Test I Information (Fall 2024) PDF

    Description

    Prepare for the ST259 Probability Test I, focusing on key concepts from Parts 1, 2, and 3 of the lecture notes. This quiz covers various probability properties such as the addition rule, Bayes' Formula, and the general multiplication rule, along with practical questions related to sample space and events. Make sure to review your formulas for a solid performance.

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