hw6 - Homework Set No 6 Probability Theory(235A Fall 2009...

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Homework Set No. 6 – Probability Theory (235A), Fall 2009 Posted: 11/3/09 — Due: 11/10/09 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) . 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. 3. Martians communicate in a binary language with two symbols, 0 and 1. A text of length n
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