MIT6_042JF10_assn10

MIT6_042JF10_assn10 - 6.042/18.062J Mathematics for...

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 6.042/18.062J Mathematics for Computer Science November 08, 2010 Tom Leighton and Marten van Dijk Problem Set 10 Problem 1. [15 points] Suppose Pr {} : S → [0 , 1] is a probability function on a sample space, S , and let B be an event such that Pr { B } > 0. Define a function Pr B {·} on outcomes w ∈ S by the rule: Pr B { w } = Pr { w } / Pr { B } if w ∈ B, (1) if w / ∈ B. (a) [7 pts] Prove that Pr B {·} is also a probability function on S according to Definition 14.4.2. (b) [8 pts] Prove that Pr B { A } = Pr { A ∩ B } Pr { B } for all A ⊆ S . Problem 2. [20 points] (a) [10 pts] Here are some handy rules for reasoning about probabilities that all follow directly from the Disjoint Sum Rule. Use Venn Diagrams, or another method, to prove them. Pr { A − B } = Pr { A } − Pr { A ∩ B } (Difference Rule) ¯ Pr A = 1 − Pr { A } (Complement Rule) Pr { A ∪ B } = Pr { A } + Pr { B } − Pr { A ∩ B } (Inclusion-Exclusion) Pr { A ∪ B } ≤ Pr { A } + Pr { B } . (2-event Union Bound) If A ⊆ B, then Pr { A } ≤ Pr { B } . (Monotonicity) 2 Problem Set 10 (b) [10 pts] Prove the following probabilistic identity, referred to as the Union...
View Full Document

This note was uploaded on 01/19/2012 for the course CS 6.042J / 1 taught by Professor Tomleighton,dr.martenvandijk during the Fall '10 term at MIT.

Page1 / 4

MIT6_042JF10_assn10 - 6.042/18.062J Mathematics for...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online