University Problem Set #8 ECE 413
of Illinois Page 1 of 1 Fall 2004
Assigned: Wednesday, October 27
Due: Wednesday, November 3
Reading: Lectures 25, 26, 27 from Lecture notes on class webpage. Chapter 5, Sections 5.4
(pp. 199-210), 5.7 (pp. 223-225), and Chapter 9, Section 9.1 (pp. 432-435) of
Ross.
Noncredit Exercises: (Do not turn these in) Self-test problems #9, 11 on pp. 236-237, #1, 2 on
p. 454 of Ross.
Problems:
53. Consider a Poisson process with arrival rate µ. Let X denote the time of the first arrival
after t = 0, and let τ denote a nonnegative real number.
(a) What is P{X > τ}?
(b) What is the probability density function of X?
(c) Let A denote the event that there are exactly four arrivals in the time interval (0, 6].
What is P(A)?
(d) Let B denote the event that there are exactly two arrivals in the interval (4, 10]. What is
P(AB)?
(e) What is the conditional probability that {X > τ} given that the event A occurred? Be sure
to give the answer for all nonnegative values of τ.
54.(a) Problem 15, p. 229, Ross.
(b) Theoretical Exercise 9, p. 233, Ross.
55. Problem 16, p. 229, Ross.
56. Problem 30, pp. 230-231, Ross.
57. X is a continuous random variable with pdf fX(u) = 0.5 exp (– |u|), –∞ < u < ∞.
(a) What is the value of P{X ≤ ln 2}?
(b) Find the conditional probability that P{|X| ≤ ln 2} given that {X ≤ ln 2}.
(c) Find the numerical value of P{cos(πX/2) < 0}.
(d) Now suppose that X denotes the voltage applied to a semiconductor diode, and that the
current Y is given by Y = eX – 1. Find the pdf of Y.
58. The random variable X has probability density function fX(u) = ⎩
⎨
⎧2(1 – u),0 ≤ u ≤ 1,
0,elsewhere.
Let Y = (1 – X)2.
(a) What is the CDF FY(v) of the random variable Y? Be sure to specify the value of FY(v)
for all v, –∞ < v < ∞.
(b) What is the pdf fY(v) of the random variable Y? Be sure to specify the value of fY(v) for
all v, –∞ < v < ∞.