20_Poisson_Est_Bayesian_Exceedances

20_Poisson_Est_Bayesian_Exceedances - CEE 597 - Lecture 20...

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 3 March 2008 Jery Stedinger Lecture 20 1 CEE 597 - Lecture 20 Poisson Parameter Estimation, Bayesian Inference, and Poisson Process plus Exceedances
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 3 March 2008 Jery Stedinger Lecture 20 2 Schedule/Learning Objectives Today’s lecture topics ar statistical/probability modeling  topics which are partly supplemental. Wednesday lecture is a review before Exam Thurs PM. From later studies, students should be able to provide:   1. justification for use of Maximum Likelihood Estimators  (MLEs) of dose-response relationships, and 2. to explain graphically or in words relationship between  values of the likelihood function, and uncertainty in model  parameters (Crump, notes, homework). 
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 3 March 2008 Jery Stedinger Lecture 20 3 Moment  λ  Estimators for Poisson Process Bias Variance Example: Babies in Tompkins County Maximum Likelihood  λ  Estimator Bayesian Inference:  prior-to-posterior  λ   uncertainty distributions Poisson Process plus exceedances Question III Annual Maximum distribution Classic distribution for maxima: EV1
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 3 March 2008 Jery Stedinger Lecture 20 4 Highway Safety Unit  Readings Next Week 1.  For Monday:  National Research Council, “55: A Decade of  Experience,” Executive Summary and Appendices in Packet.   [NRC presents a statistical analysis of a complex and  significant problem.] 2. For Wednesday, PLEASE read: Lave, “Cost of Going 55” 3. Friday start multi-media; no reading HOMEWORK #7   due Wed. March 26: AFTER Spring break.
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 3 March 2008 Jery Stedinger Lecture 20 5 Workplace fatalities – Ithaca Journal
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 3 March 2008 Jery Stedinger Lecture 20 6 Poisson Process Mathematically, a Poisson Process satisfies 3 conditions: Probability of a failure in a short interval  t equals  λ∆ t.  For small  t, probability of 2 failures within  t can be neglected.  Here  λ  is  failure rate with units of per time, or counts per time. Failure rate  λ  is constant. Number of failures in non-overlapping intervals are independent. t # arrivals in (0,T) has value k
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 3 March 2008 Jery Stedinger Lecture 20 7 Poisson Distribution The number of failures K  within a fixed  time interval t has a Poisson distribution Poisson distribution  with fixed parameter:     ν  =  λ t Probabilities: P[ K = k ]   =     ν k e - ν /k!    k = 0, 1, 2 … Moments: E[K]  =    ν   =   λ t                   Var[ K ] =  ν   =   λ t
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 3 March 2008 Jery Stedinger Lecture 20 8 How should  λ  of be estimated? Method of moments estimator
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This note was uploaded on 03/29/2009 for the course CEE 5950 taught by Professor Carr during the Fall '08 term at Cornell University (Engineering School).

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20_Poisson_Est_Bayesian_Exceedances - CEE 597 - Lecture 20...

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