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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}
n2
.
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
381
38
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View Full DocumentE[# 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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 Spring '08
 KLEINBERG
 Algorithms

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