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Lec1-1

# Lec1-1 - Stat 150 Stochastic Processes Spring 2009 Lecture...

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Stat 150 Stochastic Processes Spring 2009 Lecture 1: To be determined Lecturer: Jim Pitman Stochastic Process : Random process, evolution over time/space. Especially models for sequences of random variables: X 0 ,X 1 ,X 2 ,... Time = { 0 , 1 , 2 ,... } Random vector ( X 0 ,X 1 ,...,X N ) ( X t ,t ∈ { 0 , 1 ,...,N } ) ( X t , 0) Time = [0 , 1] In background always have Probability space: (Ω , F , P ) TRIPLE Set Ω = “set of all possible outcomes” Collection F of “events” = subsets of Ω Assume F is closed under S , T , complement, and countable set operations. If A 1 ,A 2 ,... is a sequence of events, we consider A 1 S A 2 S A 3 ... is an event; – Axiom: If the A i ’s are disjoint, A i T A j = ,i 6 = j , then P ( S i =1 A i ) = i =1 P ( A i ). Measurability. P { ω : X ( ω ) x } = F ( x ). F is called the cumulative distri- bution function of X . P ( X x ) = F ( x ). Discrete value

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Lec1-1 - Stat 150 Stochastic Processes Spring 2009 Lecture...

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