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Q6 At all times, an urn contains N balls- some white balls and some black balls. At each stage, a coin having probability p, 0

Q6 At all times, an urn contains N balls- some white balls and some black balls. At each stage, a coin having probability p, 0

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.8: Probability
Problem 32E
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Q6 At all times, an urn contains N balls- some white balls and some black balls. At each stage, a coin having probability p, 0 < p < 1, of landing heads, is flipped. If heads appear, then a ball is chosen at random from the urn and is replaced by a white ball; if tails appear, then a ball is chosen from the urn and is replaced by a black ball. Let Xn denote the number of white balls in the urn after the n stages. Q6(i.) Is {Xn, n 2 0} a Markov Chain? If so, explain why. Q6(ii.) Compute the transition probabilities Pij: (Hint: At some stage, suppose there are i white balls (and N – i black balls). Think how you can get i or i – 1 or i +1 white balls in the next stage.) Q6(iii.) Let N 2. Then the states are 0, 1, 2 and the transition probability matrix is given by 를 + 올 끌 Find the proportion of the time in each state.
Transcribed Image Text:Q6 At all times, an urn contains N balls- some white balls and some black balls. At each stage, a coin having probability p, 0 < p < 1, of landing heads, is flipped. If heads appear, then a ball is chosen at random from the urn and is replaced by a white ball; if tails appear, then a ball is chosen from the urn and is replaced by a black ball. Let Xn denote the number of white balls in the urn after the n stages. Q6(i.) Is {Xn, n 2 0} a Markov Chain? If so, explain why. Q6(ii.) Compute the transition probabilities Pij: (Hint: At some stage, suppose there are i white balls (and N – i black balls). Think how you can get i or i – 1 or i +1 white balls in the next stage.) Q6(iii.) Let N 2. Then the states are 0, 1, 2 and the transition probability matrix is given by 를 + 올 끌 Find the proportion of the time in each state.
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