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ECE313.Lecture25

# ECE313.Lecture25 - ECE 313 Probability with Engineering...

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Poisson Random Processes 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 24 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 2 of 42 Exponential random variables Exponential random variables arise in studies of waiting times, service times, etc X is called an exponential random variable with parameter λ if its pdf is given by f(u) = λ •exp(– λ u) for u ≥ 0 f(u) = 0 for u < 0 Scale parameter λ > 0 E[ X ] = 1/ λ var( X ) = 1/ λ 2
ECE 313 - Lecture 24 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 3 of 42 CDF of exponential RV X = exponential RV with parameter λ F(t) = P{ X ≤ t} = area under pdf to left of t = 1 – exp(– λ t) P{ X > t} = exp(– λ t) = complementary CDF E[ X ] = P{ X > t} dt = 1/ λ P{ X > t+ τ | X > t}= P{ X > t+ τ }/P{ X > t} = exp(– λτ ) Memoryless property of exponential RVs

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ECE 313 - Lecture 24 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 4 of 42 Gamma random variables X is called a gamma random variable with parameters (t, λ ) if its pdf is given by f(u) = λ •exp(– λ u)•( λ u) t–1 / Γ (t) for u > 0 f(u) = 0 for u ≤ 0 t > 0 is the order parameter λ > 0 is the scale parameter Γ (t) is a number whose value is the gamma function evaluated at t
ECE 313 - Lecture 24 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 5 of 42 What’s this Γ (t) stuff, anyway? Γ (t) = x t–1 •exp(–x) dx, t > 0 0 Γ (t) = (t–1)• Γ (t–1) = (t–1)•(t–2)• Γ (t–2) = … If t is an integer, Γ (t) = (t–1)! If t ≠ integer, Γ (t) = (t–1)•(t–2)•…• Γ (t– t ) where t– t is the fractional part of t For 0 < t < 1, numerical integration must be used to evaluate Γ (t) The value of Γ (t) is given by

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ECE 313 - Lecture 24 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 6 of 42 Gamma pdfs: different values of t 0 0.2 0.4 0.6 0.8 1 0 2 4 6 8 10 f(u) u λ=1 τ=1 τ=2 τ=3 τ=4 τ=5
ECE 313 - Lecture 24 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 7 of 42 Gamma pdfs: different values of λ 0 0.2 0.4 0.6 0.8 1 0 2 4 6 8 10 f(u) u t = 3 λ=1 λ=2 λ=3

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ECE 313 - Lecture 24 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 8 of 42 Mean & variance of gamma RVs If X is a gamma RV with parameters (t, λ ), then E[ X ] = t/ λ and var( X ) = t/ λ 2 A gamma RV with order parameter t = 1 is an exponential RV with parameter λ A gamma RV with order parameter t = n is called an n-Erlang random variable A gamma RV with t = n/2, λ = 1/2 is a chi-square RV with n degrees of freedom
ECE 313 - Lecture 24 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 9 of 42 An analogy The exponential RV with parameter λ is

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