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Unformatted text preview: Stochastic Processes Review: What is a Random variable? Definition: A stochastic process is a set of random variables indexed by some indices, particularly time t , and is usually denoted by X ( t ) . Some remarks: 1. Stochastic process is a mathematical model of reality. 2. Modeling usually requires somehow specifying the joint distribution ( X ( t 1 ) , ··· ,X ( t k )) or P ( X 1 ∈ A 1 , ··· ,X k ∈ A k ) 3. Values of X ( t ) are called states , the set of all possible values for X ( t ) is called the state space . The example about ’hits on a webpage’ is a typical example of stochastic process, and it has a special name: Poisson Process . 1 Poisson Process Review: What is Exponential distribution? and Poisson distribution? 1. Exponential : P ( T ≤ t ) = 1 e λt for all t ≥ where T is waiting time for rare event to happen (once). 2. Poisson : P ( X = k ) = e λ λ x /x ! where X is the number of observations of rare event during certain time period (or space). 3. pdf of Exponential distribution: f T ( t ) = λe λt for t ≥ , and λ is the rate, 1 / time. What is E ( T ) , and Var ( T ) ? What is E ( X ) and Var ( X ) ) 4. Lack of memory property for Exponential: P ( T > t + s  T > t ) = P ( T > s ) (this is key for Poisson process later) 5. Exponential race: P (min( S,T ) > t ) = P ( S > t,T > t ) = e ( λ + μ ) t if T,S independent. What about P (min( T 1 , ··· ,T n ) > t ) ? 2 Poisson process Definition: A stochastic process X ( t ) is called homogenous Poisson process with rate λ , if 1. for t > , X ( t ) takes values in { , 1 , 2 , 3 ,... } ....
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 Spring '11
 Zhou
 Probability theory, Poisson process

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