Practice Midterm9

Practice Midterm9 - Final Exam - ST 8533: Applied...

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Final Exam - ST 8533: Applied Probability Spring, 2001 May 9, 2001 Directions: For question numbers one through 15, write in the word or words that best complete the de f nition or theorem. Each blank is worth three points. No partial credit is given for an incorrect answer. 1 . Thestochast icprocess { X ( t ); t 0 } is called a Poisson process if (a) it is a counting process; (b) X (0) = 0; (c) the process has independent increments; (d) the number of events in any interval of length t has a Poisson distribution with mean λ t, t 0; that is, for all s,t 0, P { N ( t + s ) N ( s )= n } = ( e λ t ( λ t ) n n ! ,n =0 , 1 , 2 ,..., 0 , otherwise. 2. A stochastic process { X ( t ); t 0 } is said to be second-order stationary if (a) E [ X ( t )] = μ ,and (b) Cov [ X ( t ) ,X ( t + s )] = R ( s ) is a function depending only upon s and not upon t . 3. The limiting probabilities π j =l im n →∞ P n ij of a Markov chain exist only if the Markov chain is irreducible and ergodic . 4. Consider a counting process N ( t ). The process is said to have stationary increments if the number of occurrences which occur in disjoint time intervals of equal length depends only upon the length of thet imeinterva l . 5. A state i is ergodic if it is positive recurrent and aperiodic. 6. A stochastic process { X ( t ); t 0 } is called Brownian motion if 1
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(a) X (0) = 0; (b) { X ( t ); t 0 } has stationary and independent increments; and (c) for every t> 0 ,X ( t ) is normally distributed with mean 0 and variance σ 2 t . 7. Let P { X = x j } = Cb j ,j 1, where the b j are speci f ed, by C is unknown. Suppose it was of interest to estimate θ =E[ h ( X )] = X j =1 h ( x j ) P { X = x j } where h is some function which is computationally di cult. What simulation algorithm would be used in this case? Hastings-Metropolis 8. For any state i ,let f i denote the probability that, starting in state i , the process with reenter state i . If f i < 1, the process is transient . 9. Let B ( t ) represent a Brownian motion process. Then the f rst time B ( t )h itstheva lue a is called the hitting time of a . 10. A stochastic process is called a Gaussian process if X ( t 1 ) ,...,X ( t n ) has a multivariate normal distribution for all n, t 1 ,...,t n . 11. A Markov chain having initial probabilities α j equal to the limiting probabilities π j for all states j =1 , 2 ,...,n in the state space is said to be stationary . 12. A Markov chain is irreducible if there is only one class; that is, if all states communicate. 13. Let { X ( t ); t 0 } beaBrown ianmo t ionp roce ss . Thep s sfo rva s0 t 1 conditional on X ( t ) = 0 — that is, the process { X ( t ) , 0 t 1 | X (1) = 0 } —isca l leda Brownian bridge . 14. A stochastic process { X ( t ); t 0 } is said to be stationary if for all n,s,t 1 n , the random vectors X ( t 1 ) ( t n )and X ( t 1 + s ) ,...X ( t n + s )havethesamejo int distribution. 15. Suppose state i is recurrent. If the expected time until the process returns to state i is f nite, then the state is positive recurrent .
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This note was uploaded on 04/08/2008 for the course STAT 8553 taught by Professor Staff during the Spring '08 term at Columbia.

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Practice Midterm9 - Final Exam - ST 8533: Applied...

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