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UC Davis | STATS 150
8 sample documents related to STATS 150
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UC Davis STATS 150
Statistics 150: Spring 2007 April 21, 2008 0-1 1 Continuous-Time Markov Chains Consider a continuous-time stochastic process (Xt )t0 taking values in the set of nonnegative integers. We say that the process (Xt )t0 is a continuous-time Markov chain if for -
UC Davis STATS 150
Stochastic Processes Stat 150 Homework 11: Wednesday, April 30 (1) Let (Yn , Fn )n0 be a martingale. (i) Show that E [Yn ] = E [Y0 ] for all n. (ii) Show that E [Yn+m |Fn ] = Yn for all n, m 0. Due: Wednesday, May 7 (2) A supermartingale is a sequence (X -
UC Davis STATS 150
Stochastic Processes Stat 150 Homework 10: Wednesday, April 23 Due: Wednesday, April 30 (1) Consider a Yule process with X(0) = i. Given that X(t) = i + k, what is the conditional distribution of the birth times of the k individuals born in (0, t)? (2) Co -
UC Davis STATS 150
Statistics 150: Spring 2007 May 2, 2008 0-1 1 Introduction Denition 1.1. A sequence Y = cfw_Yn : n 0 of real-valued random variables is a martingale with respect to the sequence X = cfw_Xn : n 0 of random variables if, for all n 0, 1. E [|Yn |] < 2. E [Yn -
UC Davis STATS 150
Stochastic Processes Stat 150 Homework 8: Wednesday, April 9 Due: Wednesday, April 16 (1) We have n items labeled with 1, 2, . . . , n. Each day an item is requested; it is the one with the ith label with probability Pi , i 1, n Pi = 1. These item are at -
UC Davis STATS 150
1 Stationary distributions and the limit theorem Denition 1.1. The vector is called a stationary distribution of a Markov chain with matrix of transition probabilities P if has entries (j : j S) such that: (a) j 0 for all j, j j = 1, and i (b) = P, which -
UC Davis STATS 150
Statistics 150: Spring 2007 April 23, 2008 0-1 1 Limiting Probabilities If the discrete-time Markov chain with transition probabilities pij is irreducible and positive recurrent; then the limiting probabilities pj = limt Pij (t) are given by pj = j /j i i -
UC Davis STATS 150
Stochastic Processes Stat 150 Homework 9: Wednesday, April 16 Due: Wednesday, April 23 (1) Consider an irreducible, reversible, discrete-time Markov chain on the state space cfw_0, 1, 2, . . . with transition probabilities Pij and stationary probabilities
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