# 2003 wk1 - ASOC ACTL2003 Support Notes Week 1 written by...

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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 + τ- X t 1 + τ has the same distribution as X t 2- X t 1 for all t 1 ,t 2 and τ > . 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 ....
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2003 wk1 - ASOC ACTL2003 Support Notes Week 1 written by...

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