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ECE313.Lecture26

# ECE313.Lecture26 - ECE 313 Probability with Engineering...

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Gaussian Random Variables Professor Dilip V. Sarwate Department of Electrical and Computer Engineering © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign. All Rights Reserved ECE 313 Probability with Engineering Applications

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ECE 313 - Lecture 26 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 2 of 41 What is a Poisson process? Consider a sequence of random arrivals (occurrences of an event of interest) The time between two successive arrivals varies randomly: it is a random variable The inter-arrival times are independent RV arising from independent trials Observed average value of the inter- arrival times = {time of the N-th arrival}/N Observed average ≈ expected value of RV
ECE 313 - Lecture 26 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 3 of 41 Arrival rate μ Let 1/ μ denote the expected value of the inter-arrival time For large N, the total of N inter-arrival times is roughly N•(1/ μ ) = T The number of arrivals in a long time interval of duration T is roughly N = μ T μ is called the arrival rate or intensity On average, there are μ T arrivals in an interval of duration T

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ECE 313 - Lecture 26 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 4 of 41 Basic assumptions At any time instant t, at most one arrival can occur Assumptions: For small values of T P{ N (t, t+ t] = 1} = μ T P{ N (t, t+ t] = 0} = 1 – μ T The numbers of arrivals in disjoint intervals (i.e. non-overlapping intervals) of time are independent
ECE 313 - Lecture 26 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 5 of 41 P{no arrivals in (0, t]} The first arrival occurs at random time X 1 P 0 (t) = P{ no arrivals in (0, t]} = P{ X 1 > t} = μ •P 0 (t); P 0 (0) = 1 dP 0 (t) dt P 0 (t) = exp(– μ •t) for t ≥ 0 P 0 ( t) = P{ no arrivals in (0, t] = exp(– μ t) ≈ 1 – μ t for small values of t P{ X 1 > t} = exp(– μ •t) for t ≥ 0

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ECE 313 - Lecture 26 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 6 of 41 Distribution of the first arrival time The first arrival occurs at random time X 1 P{ X 1 > t} = exp(– μ •t) for t ≥ 0 Complementary CDF of X 1 is the same as that of an exponential RV with parameter μ f X 1 (t) = – derivative of complementary CDF = μ •exp(– μ •t) for t ≥ 0 X 1 is an exponential RV with parameter μ
ECE 313 - Lecture 26 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 7 of 41 P{exactly k arrivals in (0, t]} The k-th arrival occurs at random time X k P k (t) = P{exactly k arrivals in (0,t]} = μ •P k (t) + μ •P k–1 (t); P k (0) = 0 dP k (t) dt Use LaPlace transforms to solve P k (t) = exp(– μ •t) = P{ N (0, t] = k} ( μ •t) k k!

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ECE313.Lecture26 - ECE 313 Probability with Engineering...

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