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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 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 S n n = Z Ω f X 1 + · · · + X n n d P = Z 1 0 · · · Z 1 0 f 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.

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

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