randomized algorithms - 38 4/21/08 - Randomized algorithms...

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4/21/08 - Randomized algorithms Randomized Algorithms Review Ch. 13.12, read 13.1 - 6. A probability space consists of a sample set Ω (Fnite in CS 482) and a probability 0 Pr(x) 1 for every x Ω , s.t. Σ x Ω (x) = 1. Examples. ±lipping coin n times Ω = {H,T} n Pr(x) = 2 -n x Shu f ing n cards Ω = {perm’ns of {1, . .., n}} Pr(x) = 1/n! x. An event is a subset ξ Ω . Pr( ξ ) = Σ x ξ Pr(x). Ex. Event ξ = “Frst 2 coin²ips have same outcome.” ξ = {HH, TT} x {H,T} n-2 . Pr( ξ ) = 1/2. A random variable is a function Ω -> R. (X,Y) Its expectation is E[X] = Σ x Ω X(x)Pr(x) Linearity of expectation E[X+Y] = E[X] + E[Y]. Expected # of heads in n fair coin tosses. E{X] = Σ x {H,T} n (# heads in x)2 -n = Σ i=0 n i( i n )2 -n 38-1 38
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E[# heads] = Σ i=1 n E[# heads on ith coin toss] = n/2 {1 with prob 1/2 {0 with prob 1/2 E[# heads on ith] = (1/2)(1) + (1/2)(0) = 1/2. Conditional probability.
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randomized algorithms - 38 4/21/08 - Randomized algorithms...

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