Download Ecole d'Ete de Probabilites de Saint-Flour VI. 1976 by J. Hoffmann-Jörgensen, T. M. Liggett, J. Neveu, P.-L. PDF

By J. Hoffmann-Jörgensen, T. M. Liggett, J. Neveu, P.-L. Hennequin

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Extra resources for Ecole d'Ete de Probabilites de Saint-Flour VI. 1976

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If we define the random variable X k as the payoff in the kth repetition of the game, then the theoretical law of large numbers for expected value can be stated as the average payment n1 (X 1 + X 2 + · · · + X n ) over the first n repetitions of the game will converge with probability 1 to a constant as n → ∞ and this constant is equal to the expected value E(X ). 2). In many practical problems, it is helpful to interpret the expected value of a random variable as a long-term average. The law of large numbers justifies this intuitive interpretation.

4 The drunkard’s walk The drunkard’s walk is named for the drunkard exiting a pub who takes a step to the right with a probability of 12 or a step to the left with a probability of 1 . Each successive step is executed independently of the others. The following 2 questions arise: what is the probability that the drunkard will ever return to his point of origin, and what is the expected distance back to the point of origin after the drunkard has taken many steps? These questions seemingly fall into the category of pure entertainment, but, in actuality, nothing could be further from the truth.

I 20 is equal to 1. This number is equal to 10 × 10 × 18!. Thus, P(A) = 10 × 10 × 18! 263. 20! 19 × 20 The probability p that your friend will win the contest is greater than P(A) and is then, indeed, greater than 25%. Using this reasoning you will also come to the same conclusion if 100 people or even one million people write down a random number on a piece of paper and your friend allows half of the pieces to go by without choosing one. 349 of winning when the number of people participating is 20 and 100, respectively.

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