University Problem Set #8 ECE 313
of Illinois Page 1 of 2 Spring 2003
Assigned: Wednesday, March 12
Due: Wednesday, March 19
Reading: Yates and Goodman: Chapter 2.4 and 4
Reminder: No class on Friday March 14 in honor of Engineering Open House
Problems:
1. Which of the following are valid cumulative probability distribution functions (CDFs) ?
For those that are not valid CDFs, state at least one property of CDFs which is not
satisfied. For those which are valid CDFs, compute P{|X| > 0.5}.
(a) FX(u) =
0, u < 1,
2u – u2,1 ≤ u ≤ 2,
1, u > 2. (b) FX(u) =
(1/2) exp (2u), u < 0,
1– (1/4) exp (–3u), u ≥ 0.
(c) FX(u) =
(1/2) exp (2u), u ≤ 0,
1 – (1/4) exp (–3u), u > 0.
2. Let X denote the number of hours that a student works on ECE 340 each week. It is
known that X is a mixed random variable with cumulative probability distribution function
(CDF) FX(u) given by
FX(u) =
0, u < 0,
(1+u)/8, 0 ≤ u < 1,
1/2, 1 ≤ u < 2,
1/2 + u/8, 2 ≤ u < 4,
1, u ≥ 4.
Find the probability that the student
(a) works for exactly 2 hours, (b) works for more than 2 hours,
(c) works for less than 2 hours, (d) works for exactly 3 hours,
(e) works for more than 1/2 but less than 3 hours,
(f) works for more than 2 hours given that the student works at all, i.e. find P{X > 2|X > 0}.
(g) Find E[X].
3.(a) Yates and Goodman, Problem 2.5.11 on page 83. If you decided a long time ago that Y&G was
not a book to be put down lightly but one to be thrown with great force, the problem asks you to prove
that E[X] = ∑
k=0
∞ P{X > k} for a discrete random variable X that takes on nonnegative integer values only.
(b) Find P{X > k} for k = 0, 1, 2, … for a geometric random variable X with parameter p.
Substitute your answers in the formula of part (a) and give yet another derivation of the
result that E[X] = 1/p.
4. Which of the following are valid probability density functions? Assume that the functions
are zero outside the ranges specified. For those which are not valid pdfs, state at least one
property of pdfs which is not satisfied. Also, state whether there exists a constant C such
that Cf(u) is a valid pdf even though f(u) is not.
(a) f(u) = |u| for |u| < 1. (b) f(u) = 1 – |u| for |u| < 1.
(c) f(u) = ln u for 0 < u < 1, (d) f(u) = ln u for 0 < u < 2. Hint: ln u can be integrated by parts
(e) f(u) = 2u for 0 < u < 1. (f) f(u) = (2/3)(u – 1) for 0 < u < 3.
(g) f(u) = exp(–2u), 0 < u < ∞,(h) f(u) = 4 exp(–2u) – exp(–u), 0 < u < ∞.
