University Problem Set #9 ECE 313
of Illinois Page 1 of 2 Spring 2003
Assigned: Wednesday, March 19
Due: Wednesday, April 2
Reading: Yates and Goodman: Chapter 4
Reminder: Spring Break will be much more enjoyable if you go easy on the beer and heavy on
the suntan lotion instead of the other way around…
Problems:
1. [“Extra! Extra! Read all about it!”] A newsboy purchases H newspapers for c2 cents each and
sells them for c3 cents each. He can return unsold papers to the publisher for c1 cents each.
Note that c1 < c2 < c3. The daily demand X for papers is a (nonnegative) integer-valued
random variable with pmf pX(u) and CDF FX(u).
(a) What is the probability that he sells all H newspapers? Express your answer in terms of
pX(u) and also in terms of FX(u).
(b) Let Z denote the daily profit (in cents) that the newsboy makes. Write an expression for Z
in terms of X and H
(c) Use LOTUS to write an expression for his average daily profit. Your answer will depend
on H, so call the expression for the average daily profit the function g(H).
(d) The newsboy has been buying H papers for some months and making an average profit
g(H) each day. One day, he decides to buy one extra paper. What is the probability that he
can sell this extra paper? Show that he makes an average additional profit of
(c3 – c2) – (c3 – c1)FX(H) from the extra paper. Call this A(H).
(e) Show that the average additional profit A(H) = (c3 – c2) – (c3 – c1)FX(H) satisfies
…A(H–1) ≥ A(H) ≥ A(H+1) … ,
that is, on average, each extra newspaper brings in smaller extra profit than the previous
one. [Hint: FX is a non-decreasing function…] This is called the law of diminishing returns.
(f) Show that for sufficiently large H, A(H) is negative so that the newsboy loses money (on
the average) by buying too many extra papers.
(g) How many papers should he purchase to maximize his average profit?
2. X is uniformly distributed on [–1, +1].
(a) If Y = X2, what are the mean and variance of Y?
(b) If Z = g(X) where g(u) =
u2, u ≥ 0,
–u2,u < 0, use LOTUS (or the EZ method) to find E[Z]
(c) On a completely unrelated LOTUSian question, if X is a geometric random variable with
parameter 1/2, and Y = sin(πX/2), what is the value of E[Y]?
3. We return to Problem 6 of Problem Set #8. Suppose that the owner makes a gross profit
of $0.64 for each gallon of gasoline sold. Let Y denote the amount of gasoline sold per
week.
(a) How is Y related to X, the weekly demand for gasoline? (Hint: the owner cannot sell more
gasoline each week than the tank can hold!)
(b) What is the average weekly gross profit?
(c) Suppose that the owner pays $20C as weekly rent on a tank of capacity 1000C gallons.
Note that 0 ≤ C ≤ 1. (Why is a tank larger than 1000 gallons not needed?) What is the
average weekly net profit and what value of C maximizes the average weekly net profit?
4. The lifetime of a VLSI chip can be modeled as an exponential random variable with
parameter λ = –ln 0.999/week.
(a) What is the average lifetime (in weeks)? What is the median lifetime?
(b) What is the probability that the chip lasts for at least one week?
Now suppose that three identical chips are organized into a triple–modular–redundancy
(TMR) system in which we assume that the majority–logic gate cannot fail. Furthermore,
