ST259 Probability Test I Review

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

How does the multiplication rule apply to independent events?

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

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

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?

What is the complement rule according to the probability function?

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

What does the domination principle state regarding probabilities?

What is the law of total probability used for?

Which statement correctly describes independent events A and B?

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

What is the difference rule for probability functions?

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

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)

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.