ST259 Test I Information (Fall 2024) PDF

ST259 Test I Information (Fall, 2024) Page 1 of 2 Date: October 11, 2024 at 4:00 p.m. Location: LH3094 Duration: 75 min Aids allowed: Non-programming calculators are permitted; a page with potentially useful formu- lae (see the next page)...

ST259 Test I Information (Fall, 2024) Page 1 of 2 Date: October 11, 2024 at 4:00 p.m. Location: LH3094 Duration: 75 min Aids allowed: Non-programming calculators are permitted; a page with potentially useful formu- lae (see the next page) is included in the test paper Structure: Test I covers the material from Parts 1, 2, and 3 of Lecture Notes. (A) Theoretical questions (20%-25% of the test) are based on the following list: (1) Formulate and prove by using axioms of probability the following properties of the probability function: the complement rule P(A0 ) = 1 − P(A) the domination principle A ⊂ B ⇒ P(A) ≤ P(B) the addition rule P(A ∪ B) = P(A) + P(B) − P(AB) the 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 ) the difference rule P(A − B) = P(A) − P(AB) (2) Formulate and prove by using rules of probability the following properties of the conditional probability function: the complement rule P(A0 | E) = 1 − P(A | E) the domination principle A ⊂ B ⇒ P(A | E) ≤ P(B | E) (3) State and prove the general multiplication rule (for three or four events). Use the general rule to derive the multiplication rule for independent events. (4) Prove the law of total probability and use it to prove Bayes’ Formula. (5) Prove that if A and B are independent, then A and B 0 are independent. (6) Prove the fundamental probabilityPformula for a discrete random variable X with pmf p and range D: P(X ∈ A) = x∈A∩D p(x) for any A ⊂ R. (B) Practical questions (75%-80% of the test) cover the following topics from Parts 1, 2, and 3 (with equal representation of all three parts): The sample space and events; computing probabilities using properties. The classical probability; computing probabilities using binomial coefficients. The conditional probability; computing probabilities using the total probability law, the multiplication rule and Bayes’ formula ST259 Test I Information (Fall, 2024) Page 2 of 2 The independence of events; the multiplication rule for independent events. The pmf and the cdf of a discrete random variable; computing probabilities using these functions; deriving the pmf from the cdf and vice versa, plotting these func- tions. The mathematical expectation of discrete √ random2 variables and functions of discrete random variables (e.g., aX + b, |X|, X, and X ). Attn: For the complete set of typical problems please refer to lecture notes, WeBWorK and lab assignments, and recommended homework problems. Should you have any question or problems related to the lecture material or the test, please contact me. Good Luck! Potentially Useful Formulae P (E1 ∪ E2 ∪ E3 ) = P (E1 ) + P (E2 ) + P (E3 ) − P (E1 ∩ E2 ) − P (E1 ∩ E3 ) − P (E2 ∩ E3 ) + P (E1 ∩ E2 ∩ E3 )   n n! n! Ck,n = = Pk,n = k k!(n − k)! (n − k)!   n n! = n1 , n2 ,... , nr n1 ! n2 ! · · · nr ! P(A1 ∩ A2 ∩ · · · ∩ An ) = P(A1 ) · P(A2 | A1 ) · P(A3 | A1 ∩ A2 ) · · · P(An | A1 ∩ A2 ∩ · · · ∩ An−1 ) P(A | Bk )P(Bk ) P(Bk | A) = P(A | B1 )P(B1 ) + · · · + P(A | Bn )P(Bn )