# ass1 - STAT455/855 Fall 2009 Applied Stochastic Processes...

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Unformatted text preview: STAT455/855 Fall 2009 Applied Stochastic Processes Assignment #1 Due Friday, Oct.2 Starred questions are for 855 students only. 1. Ross, Chapter 3 #28 (not in 8th edition). Polyas urn model supposes that an urn initially contains r red and b blue balls. At each stage a ball is randomly selected from the urn and is then returned along with m other balls of the same color. Let X k be the number of red balls drawn in the first k selections. (a) Find E [ X 1 ]. (b) Find E [ X 2 ]. (c) Find E [ X 3 ]. (d) Conjecture the value of E [ X k ], and then verify your conjecture by a conditioning argument. (e) Give an intuitive proof for your conjecture. Hint: Number the initial r red and b blue balls, so the urn contains one type i red ball, for each i = 1 ,...,r ; as well as one type j blue ball, for each j = 1 ,...,b . Now suppose that whenever a red ball is chosen it is returned along with m others of the same type, and similarly whenever a blue ball is chosen it is returned along with m others of the same type. Now, use a symmetry argument to determine the probability that anysame type....
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## ass1 - STAT455/855 Fall 2009 Applied Stochastic Processes...

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