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6.262.Lec5

# 6.262.Lec5 - DISCRETE STOCHASTIC PROCESSES Lecture 5 Review...

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6.262 Lect. 5 2/17/2010 1 DISCRETE STOCHASTIC PROCESSES Lecture 5 Review: Bernoulli & Poisson Processes – Similarities and Differences Stationary and Independent Increments PMF’s and Densities Splitting and Merging Review: Non-Homogeneous Poisson Processes Time-Dependent Splitting Produces Non-Homogeneous Poisson Processes Example: M/G/ Queue A bit more detail on order statistics Convergence of Sequences of Random Variables Deterministic convergence of a sequence of numbers. Pointwise deterministic convergence of a sequence of functions. Sequences of random variables: three examples Two new, quite strong types of convergence Sure convergence of a sequence of random variables. Almost sure convergence and the strong law of large numbers

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6.262 Lect. 5 2/17/2010 2 Bernoulli Process (p) Poisson Process ( ) (Discrete-time) stationary & (Continuous-time) stationary & independent increments independent increments Increment Distribution (X) Geometric Exponential Number of Arrivals N(m) or N(t) to Date Binomial Poisson Time S m = X 1 + --- + X m of m-th Arrival Pascal Erlang Density λ ( ( ) ) (1 ) , 0 k m k m P N m k p p m k k = = ( ) ( ( ) ) , 0, 0 ! k t t P N t k e k t k λ λ = = 1 ( ) (1 ) , 1 1 m k m m k P S k p p k m m = = 1 ( ) , 1, 0 ( 1)! m m m t S t e f t m t m λ λ = 1 ( ) (1 ) , 1 n P X n p p n = = ( ) 1 , t 0 t P X t e λ =