Question

28. Polya's 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 Xk be the number of red balls drawn in the first k selections. (a) Find ELXi] (b) Find ELX2] (c) Find ELX3] (d) Conjecture the value of E[X], and then verify your conjecture by a condi tioning 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 any given selection is red

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Answer 1

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