Stochastic

Show that x limn1 xn has a beta distribution g r 1 g

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Unformatted text preview: all of each color. At time n we draw out a ball chosen at random. We return it to the urn and add one more of the color chosen. Let Xn be the fraction of red balls at time n. To check that Xn is a martingale note that at time n there are n + k balls, so if Rn = (n + k )Xn is the number of red balls then P (Rn+1 = Rn + 1) = Xn P (Rn+1 = Rn ) = 1 Xn 173 5.5. CONVERGENCE Letting Av = {Xn = xn , . . . X0 = x0 } we have Rn + 1 E (Xn+1 |Av ) = n+k+1 1 = n+k+1 ✓ ◆ Rn Rn Rn · + 1 n+k n+k+1 n+k Rn Rn n+k · + · = Xn n+k n+k n+k+1 Since Xn 0, Theorem 5.17 implies that Xn ! X1 . Suppose that initially there is one ball of each color. To find the distribution of X1 we note that There are 2 balls at time 0, . . . n + 1 at time n 1 so the probability the probability that red balls are drawn on the first j draws and then green balls are drawn on the next n j is 1 · · · j · 1 · · · (n j ) j !(n j )! = 2···j + 1 · j + 2···n + 1 (n + 1)! A little thought shows that each outcome with j red and n j balls drawn has the same probabi...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell.

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