Chapter 24 Approximate Max Cut
CS 573: Algorithms, Fall 2013 November 19, 2013
24.1 Normal distribution
24.1.0.1 Normal distribution – proof τ 2 =
(∫ ∞
x=−∞ exp
(
−x2 2
)
dx
)2
=
∫
(x,y)∈I R2 exp
(
−x2 + y2 2
)
dxdy Change of vars: x = r cos α, y = r sin α =
∫ 2π
α=0
∫ ∞
r=0 exp
(
−r2 2
)
- det
(∂r cos α
∂r ∂r cos α ∂α ∂r sin α ∂r ∂r sin α ∂α
)
- dr dα
=
∫ 2π
α=0
∫ ∞
r=0 exp
(
−r2 2
)
- det
(cos α
−r sin α sin α r cos α
)
- dr dα
=
∫ 2π
α=0
∫ ∞
r=0 exp
(
−r2 2
)
r dr dα =
∫ 2π
α=0
[
− exp
(
−r2 2
)]∞
r=0
dα =
∫ 2π
α=0 1 dα = 2π
24.1.0.2 Multidimensional normal distribution (A) A random variable X has normal distribution if Pr[X = x] =
1 √ 2π exp(−x2/2).
(B) X ∼ N(0, 1). (C) A vector x = (x1, . . . , xn) has d-dimensional normal distributed (i.e., v ∼ N n(0, 1) if v1, . . . , vn ˜ N(0, 1) (D) Consider a vector v ∈ I Rn, such that ∥v∥ = 1. Let x ∼ N n(0, 1). Then z = ⟨v, x⟩ has normal distribution! 1