University of Illinois Fall 2003
ECE 313: Problem Set #10
Assigned: Friday, November 7, 2003
Due: Friday, November 14, 2003
1. Consider a sphere whose radius is a random variable, R, with pdf fR(u) = 2ufor 0 < u < 1.
(a) What is the average radius of the sphere?
(b) What is the average volume of the sphere? If we call a sphere with average radius an average
sphere, then does the average sphere have the average volume?
2. Justify that the hazard rate function λ(t) as defined in the lecture is not a probability density
function.
3. An RV Xhas a hazard rate function λ(t) = at2,t≥0. Determine fX(t) and FX(t) for t≥0.
4. The lung cancer hazard rate of a t-year-old smoker, λ(t), is such that
λ(t) = 0.027 + 0.00025(t−40)2, t ≥40
Assuming that a 40-year-old smoker survives all other hazards, what is the probability that the
smoker survives to (a) age 50, and (b) age 60, without contracting lung cancer?
5. Let Xbe a continuous RV with CDF FX(u).
(a) Define another random variable Yby Y=FX(X). Show that Yis uniformly distributed
over the interval [0,1], regardless of what FX(u) is.
(b) Now let’s do the reverse: We wish to generate a continuous RV Xwith a specified distribution
(CDF) F(u), and all we are given is a uniform RV on [0,1]. Define the random variable X
by X=F−1(Y), where Y∼U[0,1]. Show that Xhas CDF F(u).
6. An analog signal Xis modeled as a standard Gaussian random variable. For some applications,
only a quantized value is used (e.g., an A/D converter converts the analog signal to one of two
possible digitized values). Let Ydenote the quantized value, with Y=αfor X > 0 and Y=−α
for X≤0, where α > 0.
(a) What is the pmf of Y?
(b) Suppose that α= 1. If the signal Xhappens to have the value 1.29, what is the error made
in representing Xby Y? What is the squared error, Z= (X−Y)2? What are the error
and squared errors for the cases of X=π/4 and X=−π/4?
(c) You have been given the task of designing the quantizer that minimizes the expected value
of the squared error, E[Z]. Your only design choice here is the value for α. Find E[Z] as a
function of α, and then find the value of αthat minimizes E[Z].
7. Determine the pdf of Y= sin X, where X∼U[−π, π].
8. (extra credit) Let X∼U[0,2]. Compute the pdf of Y if Y=g(X), with g(·) being defined as
g(x) =
0x≤0
2x0≤x≤1/2
2−2x1/2≤x≤1
0x > 1
