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By Fearn T., Brown P.J., Besbeas P.

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You would take the fraction of contests won by your friend as an estimate for the probability p of your friend winning. In order to simulate the model on the computer, you need a procedure for generating a random permutation of the numbers 1, . . , n for a given value of n. 9. 24 of Chapter 3, we come back to the best-choice problem, and you may be surprised by the solution here. 368, irrespective of the value of n. The optimal strategy is to open the first ne papers and then to choose the next paper to appear with a number higher than those contained in all of the previous papers.

This result is otherwise not in conflict 24 The law of large numbers and simulation Fig. 1. A random walk of 2,000 coin tosses. with the law of large numbers, which says that n1 × (actual number of heads in n tosses minus 12 n) goes to 0 when n → ∞. 1 that the probability distribution of the proportion of heads in n tosses becomes more and more concentrated around the 50 : 50 ratio as n increases and has the property that its spread around this ratio is on the order of √1n . 3 The arc-sine law† The random walk resulting from the repeated tossing of a fair coin is filled with surprises that clash with intuitive thinking.

M. The expected value or expectation of the random variable X is then defined by E(X ) = x1 p1 + x2 p2 + · · · + x M p M . Invoking the commonly used summation sign , we get M E(X ) = xj pj. j=1 Stating this formula in words, E(X ) is a weighted average of the possible values that X could assume, where each value is weighted with the probability that X would assume the value in question. The term “expected value” can be misleading. ” An insurance agent who tells a 40-year-old person that he/she can expect to live another 37 years naturally means that you come up with 37 more years when you multiply the possible values of the person’s future years with the corresponding probabilities and then add the products together.

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A Bayesian decision theory approach to variable selection for discrimination by Fearn T., Brown P.J., Besbeas P.

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