hw6solutions

hw6solutions - Solutions to Homework Set No. 6 Probability...

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Solutions to Homework Set No. 6 – Probability Theory (235A), Fall 2011 1. Let f : [0 , 1] R be a continuous function. Prove that Z 1 0 Z 1 0 ... Z 1 0 f ± x 1 + x 2 + ... + x n n ² dx 1 dx 2 ...dx n ---→ n →∞ f (1 / 2) . Solution. Let X 1 ,X 2 ,... be independent uniformly distributed random variables on [0 , 1]. E f ³ S n n ´ = Z Ω f ³ X 1 + ··· + X n n ´ d P = Z 1 0 ··· Z 1 0 f ³ x 1 + ··· + x n n ´ dx 1 ··· dx n where we denote by Ω the product space and by P the probability measure on Ω. By the SLLN, S n n 1 2 a.s. and so does f ( S n n ´ f ( 1 2 ) a.s. because f is continuous. Now, by the bounded convergence theorem we obtain the result. An alternative and more direct proof method involves using the Weak Law of Large Numbers by following a method similar to the proof of the Weierstrass approximation theorem using Bernstein polynomials. 2. A bowl contains n spaghetti noodles arranged in a chaotic fashion. Bob performs the following experiment: he picks two random ends of noodles from the bowl (chosen uniformly from the 2 n possible ends), ties them together, and places them back in the bowl. Then he picks at random two more ends (from the remaining 2 n - 2), ties them together and puts them back, and so on until no more loose ends are left. Let L n denote the number of spaghetti loops at the end of this process (a loop is a chain of one or more spaghettis whose ends are tied to each other to form a cycle). Compute E ( L n ) and V ( L n ). Find a sequence of numbers ( b n ) n =1 such that L n b n P ---→ n →∞ 1 , if such a sequence exists. Solution. Let X k be the random variable at k th step defined by X k = 1 if a loop is formed and X k = 0 otherwise. It is not too difficult to see that the X k ’s are independent 1
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and that P ( X k = 1) = 1 2( n - k )+1 , P ( X k = 0) = n - k 2( n - k )+1 . Since the process ends after
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This note was uploaded on 11/13/2011 for the course STAT 235A taught by Professor Danromik during the Fall '11 term at UC Davis.

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hw6solutions - Solutions to Homework Set No. 6 Probability...

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