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Mathematics of Operational Research Example Sheet 2 R. Weber ...
www.statslab.cam.ac.uk/~rrw1/mor/examples2.pdf2 Dec 2015: maximizesi,tj. { i. si j. tj : αiβj si tj 0, i, j}. -
Intersection and mixing times for reversible chains Yuval Peres∗ ...
www.statslab.cam.ac.uk/~ps422/intersection-mixing.pdf7 Jan 2015: i=0. tj=0 1(Xi = Yj). count the number of intersections up to time t. ... Qt =ti=0. tj=0. pij(x,x). Using the spectral theorem together with transitivity, we obtain. -
21 Paper 4, Section I 9H Markov ChainsLet X0, ...
www.statslab.cam.ac.uk/~rrw1/markov/MarkovChainTriposQuestions.pdf17 Sep 2015: Assume p > q. Let Tj = inf{n > 1 : Xn = j} if this is finite, and Tj = otherwise. ... Let Ti = inf{n > 1 : Xn = i}. For each i 6= j letqij = P(Tj < Ti | X0 = i) and mij = E(Tj | X0 = i). -
41 Paper 1, Section I 7H StatisticsSuppose that X1, ...
www.statslab.cam.ac.uk/~rrw1/stats/StatisticsTriposQuestions.pdf17 Sep 2015: for j 6= i. Prove that the random vectors Yj AjX are independent, and thatY (Y T1 ,. , Y TJ )T has a multivariate normal distribution.[ Hint: Random vectors are independent if -
19 Paper 2, Section I 3F ProbabilityLet U be ...
www.statslab.cam.ac.uk/~rrw1/prob/ProbabilityTriposQuestions.pdf17 Sep 2015: Find the probability-generating functions of the random variables Hj and Tj.
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