This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: ASOC ACTL2003 Support Notes: Week 1 written by Tim Yip, Andy Wong and Andrew Teh Stochastic Processes: A stochastic process { X ( t ) ,t T } is a collection of random variables  for each t , X ( t ) is a random variable. T is the index set . Can be discrete (i.e. { , 1 , 2 ,... } ) or continuous. X ( t ) is the state of the process at time t . Can be discrete or continuous. The state space of the process is the set of values that X ( t ) can take. De noted by S . A sample path of a stochastic process is a record of how the process evolved in one particular case Increments A stochastic process has independent increments if the random vari ables X t ,X t 1 X t ,X t 2 X t 1 ,...,X t n X t n 1 are independent for all t 1 < t 2 < ... < t n . A stochastic process has stationary increments if X t 2 +  X t 1 + has the same distribution as X t 2 X t 1 for all t 1 ,t 2 and > . 1 Markov Chains Markov Processes are stochastic processes that have the property that the future state X t n only depends on the present state X t n 1 . Mathematically, P ( X t n = x n  X t 1 = x 1 ,X t 2 = x 2 ,...,X t n 1 = x n 1 ) = P ( X t n = x n  X t n 1 = x n 1 ) A Markov Chain is a Markov process that has a discrete index set T and a discrete state X n ....
View Full
Document
 Three '11
 kim

Click to edit the document details