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Poisson-beamer

# Poisson-beamer - Reminders About the Exponential...

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Reminders About the Exponential Distribution Poisson Process Properties of the Poisson Process Introductory Engineering Stochastic Processes, ORIE 3510 Instructor: Mark E. Lewis, Associate Professor School of Operations Research and Information Engineering Cornell University Disclaimer : Notes are only meant as a lecture supplement not substitute! 1/ 24

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Reminders About the Exponential Distribution Poisson Process Properties of the Poisson Process Some Properties of Exponentials Simple Examples Sums of iid Exponentials Reminders About The Exponential Distribution Recall that the exponential distribution is the only continuous distribution with the Memoryless Property . That is, if X Exp ( λ ) then P ( X > t + s | X > s ) = P ( X > t ) If a machine lives for s time units, the probability of it living for t more time units is the same as it living for t units (starting from time zero) Suppose X Exp ( λ ) and Y Exp ( μ ) then P ( X < Y ) = λ λ + μ . This tells us that if two exponential events are competing, the probability that the “ X ” event occurs first is given by the above expression. Suppose again that X Exp ( λ ) and Y Exp ( μ ) then min { X , Y } ∼ Exp ( λ + μ ). If two exponential events are competing, the time until one of the events occurs is exponentially distributed with rate λ + μ . 2/ 24
Reminders About the Exponential Distribution Poisson Process Properties of the Poisson Process Some Properties of Exponentials Simple Examples Sums of iid Exponentials Reminders About The Exponential Distribution Recall that the exponential distribution is the only continuous distribution with the Memoryless Property . That is, if X Exp ( λ ) then P ( X > t + s | X > s ) = P ( X > t ) If a machine lives for s time units, the probability of it living for t more time units is the same as it living for t units (starting from time zero) Suppose X Exp ( λ ) and Y Exp ( μ ) then P ( X < Y ) = λ λ + μ . This tells us that if two exponential events are competing, the probability that the “ X ” event occurs first is given by the above expression. Suppose again that X Exp ( λ ) and Y Exp ( μ ) then min { X , Y } ∼ Exp ( λ + μ ). If two exponential events are competing, the time until one of the events occurs is exponentially distributed with rate λ + μ . 2/ 24

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Reminders About the Exponential Distribution Poisson Process Properties of the Poisson Process Some Properties of Exponentials Simple Examples Sums of iid Exponentials Reminders About The Exponential Distribution Recall that the exponential distribution is the only continuous distribution with the Memoryless Property . That is, if X Exp ( λ ) then P ( X > t + s | X > s ) = P ( X > t ) If a machine lives for s time units, the probability of it living for t more time units is the same as it living for t units (starting from time zero) Suppose X Exp ( λ ) and Y Exp ( μ ) then P ( X < Y ) = λ λ + μ .
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Poisson-beamer - Reminders About the Exponential...

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