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The fortune Xn of a gambler evolves as Xn = ZnXnu22121, where the Z.pdf
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The fortune Xn of a gambler evolves as Xn = ZnXnu22121, where the Z.pdf

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The fortune Xn of a gambler evolves as Xn = ZnXn%u22121, where the Zn are independent identically distributed random variables with PMF pZ(z) = 1/3, for z = 3, 2/3, for z = 1/3. Assume that X0 = 1. (a) Show that the expected fortune E[Xn] converges to infinity as n increases. (b) Show that the actual fortune Xn converges to 0 with probability 1. Solution.

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The fortune Xn of a gambler evolves as Xn = ZnXnu22121, where the Z.pdf

  1. The fortune Xn of a gambler evolves as Xn = ZnXn%u22121, where the Zn are independent identically distributed random variables with PMF pZ(z) = 1/3, for z = 3, 2/3, for z = 1/3. Assume that X0 = 1. (a) Show that the expected fortune E[Xn] converges to infinity as n increases. (b) Show that the actual fortune Xn converges to 0 with probability 1. Solution
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