University of Illinois Fall 2006
ECE 413: Problem Set 8
Due: Wednesday October 25 at the beginning of class.
Reading: Ross, Chapters 3, 4, 5, and the notes on decision-making on the class web page
Reminder: No class on Friday October 27
Cancelled class will be made up on Monday October 30, 7-8 pm
This Problem Set contains five problems
1. The number of α-particles emitted by a source during a unit time interval can be modeled as a
Poisson random variable Xwith parameter λ. The α-particles are detected by an (imperfect)
Geiger counter which detects each particle with probability p < 1. Let Ydenote the number
of particles detected by the Geiger counter. Each detection can be considered to be an
independent event, and so the conditional pmf of Ygiven that X=nis a binomial pmf with
parameters (n, p). Thus,
pY|X =n(k|X =n) = P{Y =k|X =n}=µn
k¶pk(1 −p)n−k,for 0 ≤k≤n.
(a) What is the conditional mean of Ygiven the event {X =n}?
(b) From the result of part (a), find the unconditional mean of Y.
(c) What is the unconditional pmf of Y? [Hint: if Y=k, then Xmust have taken on some
value ≥k;Xcannot possibly be smaller than k.]
(d) What is the conditional pmf of Xgiven that Y=k?
(e) What is the conditional mean of Xgiven that Y=k?
(f) We can observe the Geiger counter reading Y, but we wish to know the value of X. If the
Geiger counter reading is k, i.e. the event {Y =k}is observed, what is the maximum
likelihood estimate of X?
2. The Senate of a certain country has 100 members consisting of 43 Conservative Republicans,
21 Conservative Democrats, 12 Liberal Republicans, and 24 Liberal Democrats. Before each
vote, the groups caucus separately. Each group decides independently of the other groups
whether to support or oppose the motion. All members of the group then vote in accordance
with the caucus decision. If you believe that this is the way politics works, I have this beautiful skyscraper on Wacker Drive
in Chicago that I am willing to sell to you at a real bargain price .. .
(a) Let A, B, C , and Drespectively denote the events that the four groups vote to eliminate
all income taxes on capital gains. Suppose that the probabilities of these independent
events are P(A) = 0.9, P (B) = 0.6, P (C) = 0.5 and P(D) = 0.2. What is the probability
that the bill passes?
(b) The President vetoes the bill as a budget-breaker. Let E, F, G, and Hrespectively denote
the independent events that the four groups support the motion to override the veto. If
these events have probabilities P(E)=0.99, P (F) = 0.4, P (G)=0.6, and P(H) = 0.1,
what is the probability that the motion to override the veto passes ?
Political innocents are reminded that a simple majority (51 or more votes) is required to pass a
bill, and a two-thirds majority (67 or more votes) to override a veto.
3. The dice game of craps begins with the player (called the shooter) rolling two fair dice. If the
sum is 2, 3, or 12, the shooter loses the game. If the sum is a 7 or 11, the shooter wins the
game. If the sum is any of 4, 5, 6, 8, 9, 10, then the shooter has neither won nor lost (as yet).
The number rolled is called the shooter’s point, and what happens next is described in parts
(b) and (c) below.
