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Add the probability that all three bits are flipped to the probability that exactly two bits are flipped. Sum rule method. Using the sum rule, compute the marginal probability that r takes on each of the eight possible values, P (r). ] Then compute the posterior probability of s for each of the eight values of r. ] Notice that some of the inferred bits are better determined than others. From the posterior probability P (s | r) you can read out the case-by-case error probability, the probability that the more probable hypothesis is not correct, P (error | r).

This is a strictly convex function. We notice that the equality E [f (x)] = f (E[x]) holds, therefore x is a constant, and the three lengths must all be equal. The area of the largest square is 100 m 2 . 49) then f (x) has its maximum value at that point. The converse does not hold: if a concave f (x) is maximized at some x it is not necessarily true that the gradient ∇f (x) is equal to zero there. For example, f (x) = −|x| is maximized at x = 0 where its derivative is undefined; and f (p) = log(p), for a probability p ∈ (0, 1), is maximized on the boundary of the range, at p = 1, where the gradient df (p)/dp = 1.

Another version of this exercise refers to Fred waiting for a bus at a bus-stop in Poissonville where buses arrive independently at random (a Poisson process), with, on average, one bus every six minutes. What is the average wait for a bus, after Fred arrives at the stop? ] So what is the time between the two buses, the one that Fred just missed, and the one that he catches? ] Explain the apparent paradox. Note the contrast with the situation in Clockville, where the buses are spaced exactly 6 minutes apart.

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A Hacker's Revenge


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