University Problem Set #4 ECE 313
of Illinois Page 1 of 3 Fall 2001
Assigned: Wednesday, September 12
Due: Wednesday, September 19
Reading: Ross, Chapter 4.1, 4.3â4.5, 4 .7, and Chapter 3
Noncredit Exercises: Chapter 4, Problems: 34, 35-38, 43, 48; Theoretical Exercises: 9, 13, 15
Problems:
1. The prospectus of GoGoDotCom Inc., an investment management service, states that their
goal is to double the value of their clientsâ investments in a week via day trading of Internet
stocks. (The Securities and Exchange Commission insists, as usual, that a disclaimer be
included that there is no guarantee that the goal will be met.) The TV commercials proclaim
âOn average, our clients triple their money in five weeks!â You decide to invest $32 (hey,
you are a student on a tight budget) with GoGoDotCom Inc. for a period of five weeks.
Let X denote the value (in dollars) of your investment at the end of this period.
Now suppose that GoGoDotCom Inc.has a 50% chance of doubling your investment and
50% chance of losing half your investment. That is, if you invest $C with them, then, a
week later, your investment is equally likely to be worth $2C or $C/2. Assume that each
weekâs performance is an independent trial that doubles or halves the value that your
investment had at the beginning of the week.
(a) What are the possible values of X?
(b) What is the pmf of the random variable X?
(c) What is the expected value of X? Is the TV commercial an accurate statement?
(d) What is the probability that you will lose money on this investment? i.e. find P{X < 32}.
(e) An investment of $32,000 with GoGoDotCom Inc. would be worth $1000X in five
weeks. Assuming that you have the money, would you be willing to make such an
investment? Why or why not? Would you be willing to borrow the money from your
parents to make the investment? How about borrowing the money from a loan shark?
2. Let X denote a binomial random variable with parameters (N, p). What is the probability
that X is an even integer? Remember that 0 is an even integer.
[Hint: What is (x+y)N + (x â y)N?]
3. There are N multiple-choice questions on a certain examination. A student knows the
answer to K of these and marks the answer sheet accordingly. For the remaining N â K
questions, the student guesses randomly among the five choices. The examiner can easily
determine C, the number of correct answers on the answer sheet, but is more interested in
estimating the value of K, since K is a better measure of the studentâs knowledge than C.
(Educators like to nitpick about such subtle differences!). Note that the number of wrong
answers can be modeled as a binomial random variable W with parameters (N â K, 0.8).
(a) The examiner notes that n questions have been answered incorrectly by the student, i.e. the
event {W = n} is observed. Write an expression for P{W = n} in terms of N, K, and n.
(b) Obviously, 0 †K †Nân. Now, use the method used in the proof of Prop. 7.1, Chapter 4
of Ross to show that of all possible assumptions K = 0, K = 1, K = 2, ⊠, K = Nân that
the examiner might make, the assumption that K is the largest integer not exceeding
N â 1.25n + 1, i.e. estimating K as ^
K = ïŁ°N â 1.25n + 1ïŁ» maximizes P{W = n}. ^
K is the
called the maximum-likelihood estimate of K. Find the numerical value of ^
K for the case
N = 100 and n = 8.
(c) Since C = N â n, examiners generally subtract one-fourth of the wrong answers from C
and estimate the value of K as ~
K = C â ïŁ°0.25nïŁ» â N â 1.25n. This is called applying the
guessing penalty, and it can hurt scores slightly in the sense that the examinerâs estimate ~
K
might be smaller than the maximum-likelihood estimate ^
K found in part (b).
Compare ^
K and ~
K for the case N = 100 and n = 8.
Compare ^
K and ~
K for the case N = 100 and n = 10.
(d) If N = 100 and K = 90, which of the events {W = 0}, {W = 1}, ⊠, {W = 10} has the
largest probability? (Hint: See Proposition 7.1, p. 150 of Ross). Suppose that this largest
