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Main points of this exam paper are: Conditionally, Independent, Events, Random Variable, Two Random, Joint Distribution, Nonin Nite Variance
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Fall 2009: EECS126 Final No Collaboration Permitted. Two sheets of notes permitted. Turn in with your exam. Be clear and precise in your answers Write your name and student ID number on every sheet. Come to the front if you have a question. 110 points is a very good score. There are more points than that on the exam. So look over the entire exam. Some points at the end may be easier to get than others earlier in the exam.
Problem 1.1 (36pts) True or False. Prove or show a counterexample:
a. 12pts. Let A 1 , A 2 , A 3 be events with 0 < P (A 3 ) < 1. Suppose A 1 and A 2 are conditionally independent given A 3 and are also conditionally indepen- dent given Ac 3. Then A 1 and A 2 must be independent.
c. 12pts. X and Y are two random variables with some joint distribution and each one individually has a well defined positive noninfinite variance. Then the mean-squared estimation error random variable X˜ = (X − E[X|Y ]) is independent of Y.
Problem 1.2 (45pts) Markov Fun (the parts of this problem are not con- nected to each other, so do them in any order)
a. 25pts. You have a server that can be in one of two states: “busy” or “idle.” Assume that the states evolve in discrete-time in a Markov fashion. Nature dictates that the probability of going from “idle” to “busy” is 9
b. 4pts. Draw a Markov chain (and label all the transition probabilities) whose probability transition matrix has eigenvalues +1 and 0.
c. 4pts. Draw a Markov chain (and label all the transition probabilities) whose probability transition matrix has eigenvalues +1 and -1.
d. 4pts. Draw a Markov chain (and label all the transition probabilities) whose probability transition matrix has eigenvalues +1 and +1.
e. 4pts. Draw a Markov chain (and label all the transition probabilities) whose probability transition matrix has eigenvalues +1 and −^15.
f. 4pts. Draw a Markov chain (and label all the transition probabilities) with two distinct recurrent classes and at least one transient state.
Problem 1.4 (15 pts.) Somebody claims that it is sufficient to have just 2 bathrooms on an airplane that carries 100 people on it, each of which wants to go to the bathroom independently of the others with some probability q. You probe this person about what does he mean by “sufficient” and he says that he means that less than 10% of the time will someone have to wait for an empty bathroom assuming that time is slotted and the time required to finish using the bathroom is 1 slot long. What can you conclude about q? Give your analysis and explain how you are modeling the problem using probability.
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Problem 1.6 (25 pts.) X and Y are jointly Gaussian random variables. We know that E[X] = E[Y ] = 0 and E[X^2 ] = 1 and E[Y 2 ] = 2. Further- more, E[XY ] = −^12. Suppose that I tell you that X, Y are both obtainable as linear combinations of independent standard (zero mean and unit variance) Gaussian random variables W, V. Do you have enough information to uniquely write X and Y as linear combinations of W, V? If so, prove it and give these unique linear combinations. If not, give at least two distinct linear combinations that are both compatible with the specified information.
