class02-2-handouts

# class02-2-handouts - PSTAT 120B Probability Statistics...

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PSTAT 120B - Probability & Statistics Class # 02-2- Central limit theorem Jarad Niemi University of California, Santa Barbara 7 April 2010 Jarad Niemi (UCSB) Central limit theorem 7 April 2010 1 / 19

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Class overview Announcements Announcements Homework 1 is returned in class, come up afterwards to get yours A minimal solution guide will be made available Complete your answers, e.g. if the mgf is e ( λ 1 + λ 2 )( e t - 1) then the random variable is a Poisson with mean λ 1 + λ 2 . I still haven’t made corrections to Monday’s slides, but I plan to Society of actuaries Open House (next slide) Jarad Niemi (UCSB) Central limit theorem 7 April 2010 2 / 19
Class overview Announcements Society of actuaries - open house Then you should consider a career as an actuary. Join us to learn more about one of today’s best careers. WHAT: Informational presentation and open house All interested students are welcome DINNER WILL BE PROVIDED. WHEN: Monday, April 12, 2010 4-6 p.m. WHERE: HSSB 1174 Please RSVP at: [email protected] For more information, contact or visit www.RiskisOpportunity.net Jarad Niemi (UCSB) Central limit theorem 7 April 2010 3 / 19

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Class overview Day’s Goals Goals Central limit theorems For averages For sums Beta distribution Continuity correction Jarad Niemi (UCSB) Central limit theorem 7 April 2010 4 / 19
Central limit theorem Proportion Suppose Y 1 , Y 2 , . . . , Y n iid Ber(0 . 5). What is the distribution for Y n = 1 n n i =1 Y i ? n = 5 Y p ( Y 29 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 n = 15 Y p ( Y 29 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 n = 25 Y p ( Y 29 0.0 0.2 0.4 0.6 0.8 1.0 0 1 2 3 4 Jarad Niemi (UCSB) Central limit theorem 7 April 2010 5 / 19

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Central limit theorem CLT Central Limit Theorem Let Y 1 , Y 2 , . . . , Y n be independent and identically distribution random variables with E ( Y i ) = μ and V ( Y i ) = σ 2 <
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