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EE 126, Spring 2006, Midterm 1
Problem 1: (14 points)
Since there is no direct flight from San Diego (S) to New York (N), every time Alice wants to go to the New York, she has to stop in either Chicago (C) or Denver (D). Due to bad weather conditions, both the flights from S to C and the flights from C to N have independently a delay of 1 hour with probability p. Similarly, at Denver airport, both incoming and outgoing flights are independently
subject to a 2 hour delay with probability q. On any given occasion, Alice chooses randomly between
the Chicago or Denver routes with equal probability.
(a) (2pt) What is the average total delay (across both legs of the overall trip) that she experiences in going from S to N?
(b) (3pt) Suppose Alice arrives at N with a delay of two hours. What is the probability that she flew through C?
(c) (3pt) Suppose that Alice wants to maximize the probability that she arrives in New York with a total
delay < 2 hours. Under what conditions on p and q is going via Chicago a better choice than going via
Denver?
(d) (3pt) Suppose now that Alice always flies through C. On average, how many trips does she make before experiencing a 2 hour delay?
(e) (3pt) Suppose now that the flight between S and D is known to be delayed, but Alice still randomly flies either via C or D with equal probability. With what delay should she expect to arrive at N?
Problem 2: (13 points)
We transmit a bit of information which is 0 with probability 1 - P and 1 with p. Because of noise on the channel, each transmitted bit is received correctly with probability 1 -E.
(a) (2pt) Suppose we observe a "1" at the output. Find the conditional probability PI that the transmitted bit is a "1".
(b) (4pt) Suppose that we transmit the same information bit n times over the channel. Calculate the probability that the information bit is a "1" given that you have observed n "l"s at the output. What happens when n grows? Does it make sense intuitively?
(c) (3pt) For this part of the problem, we suppose that we transmit the symbol "1" a total of n times over the channel. At the output of the channel, suppose that we observe the symbol "1" three times in the n received bits, and that we observe a "1" at the n-th transmission. Given these facts, what is the probability that the k-th received bit is a "I"?
(d) (4pt) Now let's go back to the situation in part (a)-that is, some unknown bit is transmitted over the channel, and the received bit is a "1". Suppose in addition that the same information bit is transmitted a second time, and you again receive another "1". We want to find a recursive formula to update PI to get P2, the conditional probability that the transmitted bit is a "1" given that we have observed two "l"s at the output of the channel. Show that the update can be written as
