CS 530: Geometric and Probabilistic Methods in
Computer Science
Homework 5 (Fall ’08)
1. Consider the following advection-diffusion partial differential equation:
∂P
∂t=−C∂P
∂x+D∂2P
∂x2.
(a) Give finite difference approximations for ∂P
∂t|x,t,∂P
∂x|x,t, and ∂2P
∂x2|x,t.
(b) Give an expression for P(x,t+∆t)in terms of P(x,t),P(x+∆x,t), and
P(x−∆x,t).
2. A bivariate Gaussian random variable, x=x0x1T, has the following
p.d.f.:
f(x0,x1) = (a2−b2)1
2
2πexp−1
2[ax0x0+2bx0x1+ax1x1].
(a) Give the matrix Wwhich will decorrelate the components of x.
(b) Let u=Wx. Give an expression for g(u0,u1), the p.d.f. for the bivari-
ate Gaussian random variable, u=u0u1T.
3. Let Sr(λ),Sg(λ), and Sb(λ)be the spectral sensitivity functions of the red,
green, and blue cones of the human retina. Let C(λ)be the spectral distri-
bution of a sunlit daffodil. Define a system of linear equations, which when
solved, gives the amounts, Vr(C),Vg(C), and Vb(C), of the three CIE stan-
dard primary sources, δ(λ−700),δ(λ−546), and δ(λ−436), necessary to
reproduce the color of the sunlit daffodil.
4. The n-th moment of Ψis defined to be Mn{Ψ}=R∞
−∞tnΨ(t)dt. Let f(t) =
e−πt2,f0(t) = −2πte−πt2, and f00(t) = 2πe−πt2(2πt2−1). Prove the follow-
ing:
(a) M0{f0}=0.
(b) M0{f00}=M1{f00}=0.
5. The six vectors, f1=cos(π/3)sin(π/3)T,f2=cos(π/3)−sin(π/3)T,
f3=−1 0 T,f4=−cos(π/3)−sin(π/3)T,f5=−cos(π/3)sin(π/3)T,
and f6=1 0 Tform a frame Ffor R2. Draw the frame.
