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17 Jun 2019: Assume p > q. Let Tj = inf{n > 1 : Xn = j} if this is finite, and Tj = otherwise.

17 Jun 2019: Let Ti = inf{n > 1 : Xn = i}. For each i 6= j letqij = P(Tj < Ti | X0 = i) and mij = E(Tj | X0 = i).

17 Jun 2019: Assume p > q. Let Tj = inf{n > 1 : Xn = j} if this is finite, and Tj = otherwise.

17 Jun 2019: Find the probability-generating functions of the random variables Hj and Tj.

17 Jun 2019: 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

17 Jun 2019: 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).

17 Jun 2019: Show that if g(s, x) is a real-valued function of realvariables s, x, and Tj are the jump times of (Nt, t > 0) then. ... E[exp. {θ. Nt. j=1. g(Tj, Xj). }]= exp. {λ. t. 0(E(eθg(s,X)) 1)ds. },.

17 Jun 2019: ci Πmj=1 taijj. subject to tj > 0 j = 1,.

17 Jun 2019: Find the probability-generating functions of the random variables Hj and Tj.

17 Jun 2019: MATHEMATICAL TRIPOS Part IB 2009. List of Courses. Analysis II. Complex Analysis. Complex Analysis or Complex Methods. Complex Methods. Electromagnetism. Fluid Dynamics. Geometry. Groups, Rings and Modules. Linear Algebra. Markov Chains.

17 Jun 2019: Show that if g(s, x) is a real-valued function of realvariables s, x, and Tj are the jump times of (Nt, t > 0) then. ... E[exp. {θ. Nt. j=1. g(Tj, Xj). }]= exp. {λ. t. 0(E(eθg(s,X)) 1)ds. },.

17 Jun 2019: minimise g(t) =n. i=1. ci Πmj=1 taijj. subject to tj > 0 j = 1,.

19 Jun 2019: c) tj and tk are times of consecutive events (tj < tk) in a relative survival dataset. ... Describe an example of a situation where. F̂E (tK ) might be greater than F̂E (tj ). END OF PAPER. Biostatistics.

19 Jun 2019: The variable tj (j = 1,. , 3) records the time interval under observationand takes the values 2, 4 or 6 months. ... logE(Yij|zi; xi; tj ). 1 E(Yij|zi; xi; tj )= α φZi βT xi δtj , (i = 1,. ,

19 Jun 2019: dependent toll Tj(yj ), and suppose each user selects a route in an attempt to mini-. ... Tj () so that the equilibrium flow pattern minimizes the average delay in the network.

19 Jun 2019: where Si1, Si2 correspond respectively to the swimming times of the strong swimmer andthe not so strong swimmer in the ith pair, for 1 6 i 6 3, and Tj corresponds ... a) Show that for j = 4,. 8, Tj has probability density function.

19 Jun 2019: connected to nodes ti and tj with arcs of infinite capacity, and each nodeti is connected to a node s′ with an arc of capacity w wi 1 and w = wn.

19 Jun 2019: Xj = min(Tj , cj ), for 1 j n. extending the notation of (b) in the natural way.

19 Jun 2019: Theeffects of recombination in the segment may be ignored. The time for which the samplehas j distinct ancestors is denoted by Tj , j = 2, 3,. ,

19 Jun 2019: Let uji be the approximation at xi and tj ,i = 1,. ,