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Unformatted text preview: PSTAT 120B  Probability & Statistics Class # 041 Midterm Review I Jarad Niemi University of California, Santa Barbara 19 April 2010 Jarad Niemi (UCSB) Midterm Review I 19 April 2010 1 / 26 Class overview Announcements Announcements Homework Homework 3 due today by 4pm in South Hall 5521 Midterm 1 next Friday (23 April) On material covered through today and hw3 Part multiple choice and part “short” answer Jarad Niemi (UCSB) Midterm Review I 19 April 2010 2 / 26 Class overview Goals Applied examples Happy Birthday Cakes Predicting UCSB presidential elections Jarad Niemi (UCSB) Midterm Review I 19 April 2010 3 / 26 Happy Birthday Cakes Suppose you are a COSTCO bakery manager and are in charge of deciding how many ‘Happy Birthday!’ cakes should be baked every day. How are you going to make this decision? Guess Estimate ∼ 55 , 000 people live in Goleta 150 birthdays a day maybe 10% buy cakes at COSTCO so 15 cakes Collect data Average is 15.5 cakes Model it! Jarad Niemi (UCSB) Midterm Review I 19 April 2010 4 / 26 Happy Birthday Cakes A statistical model for cake purchases Identify the random variable and give it a letter X : Number of cakes purchased tomorrow Select a distribution family, e.g. normal, binomial, Poisson, geometric and identify the parameters X ∼ Po ( λ ) λ : average cakes purchased per day which is unknown Collect data How many cakes should you bake while collecting data? Suppose we record the # of cakes purchased for n days. Call these X 1 , X 2 ,..., X n . Are they independent? Are they conditionally independent given λ ? Jarad Niemi (UCSB) Midterm Review I 19 April 2010 5 / 26 Happy Birthday Cakes Estimators for average number of cakes Suppose X 1 , X 2 ,..., X n iid ∼ Po ( λ ). Find an estimator for λ MOM: E [ X ] = λ set = x = ⇒ ˆ λ = x MLE: L ( λ ) = p ( x 1 , x 2 ,..., x n  λ ) ind = n Y i =1 p ( x i  λ ) = n Y i =1 e λ λ y i y i ! = e n λ λ ∑ n i =1 x i Q n i =1 x i ! ‘ ( λ ) = n λ + n X i =1 x i log λ log n Y i =1 x i ! ! d d λ ‘ ( λ ) = n + 1 λ n X i =1 x i set = 0 = ⇒ ˆ λ = x Jarad Niemi (UCSB) Midterm Review I 19 April 2010 6 / 26 Happy Birthday Cakes Simulated data You collect data for a week and get the following data: 21 18 21 15 17 13 13 So the MOM and MLE are both 15.57143. What does this mean? How many cakes are going to be purchased tomorrow? X=15.57143? X ∼ Po (15 . 57143)? So how many cakes should you bake?...
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This note was uploaded on 01/10/2011 for the course STAT 120B taught by Professor Bennett during the Fall '09 term at UCSB.
 Fall '09
 Bennett
 Statistics, Probability

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