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Extra resources for A Hacker's Revenge
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.
A Hacker's Revenge