University of Illinois Fall 2008
ECE 313: Problem Set 8
Due: Wednesday, October 29, at 4 p.m.
Reading: Ross, Chapters 4 and 5.
Non-credit exercises: Ross, Chapter 5, Problems 24,25,30, and 35.
This Problem Set contains five problems
1. Which of the following functions F(u) are valid CDFs? For those that are valid CDFs,
compute the probability that the absolute value of the random variable exceeds 0.5.
(a) F(u) =
0u < 0,
u2,0≤u < 1,
1, u ≥1.
(b) F(u) =
0u < 1,
2u−u2,1≤u≤2,
1, u > 2.
(c) F(u) = ½1
2exp(2u)u≤0,
1−1
4exp(−3u), u > 0,(d) F(u) = ½1
2exp(2u)u < 0,
1−1
4exp(−3u), u ≥0,
2. The number of hours that a student spends on ECE 440 homework is a random variable X
with CDF
FX(u) =
0, u < 0,
(1 + u)/8,0≤u < 1,
1/2,1≤u < 2,
(4 + u)/8,2≤u < 4,
1, u ≥4.
Note that this is a mixed random variable: it takes on some values with nonzero probability
(like a discrete random variable) but also takes on all values in intervals of the real line (like
a continuous random variable).
(a) Find P{X = 2},P{X <2},P{X >2},P{1≤ X ≤ 3}, and P{X >2| X >0}.
(b) Find E[X].
3. Nine functions f(u) are shown below. Note that in each case, f(u) = 0 for all unot in the
interval specified. In each case,
•determine whether f(u) is a valid probability density function (pdf).
•If f(u) is not a valid pdf, determine if there exists a constant Csuch that C·f(u) is a
valid pdf.
(a) f(u) = 2u, 0< u < 1. (b) f(u) = |u|,|u|<1
2
(c) f(u) = 1 − |u|,|u|<1,(d) f(u) = ln u, 0< u < 1. Hint: ln ucan be integrated by parts.
(e) f(u) = ln u, 0< u < 2,(f) f(u) = 2
3(u−1),0< u < 3,
(g) f(u) = exp(−2u), u > 0. (h) f(u) = 4 exp(−2u)−exp(−u), u > 0,
(i) f(u) = exp(−|u|),|u|<1,
4. Problem 11 on page 248 of Ross.
5. Problem 12 on page 248 of Ross.