MIT6_042JF10_assn10

MIT6_042JF10_assn10 - 6.042/18.062J Mathematics for...

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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...
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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.

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MIT6_042JF10_assn10 - 6.042/18.062J Mathematics for...

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