class04-1-handouts

class04-1-handouts - PSTAT 120B Probability Statistics...

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PSTAT 120B - Probability & Statistics Class # 04-1- Midterm Review I Jarad Niemi University of California, Santa Barbara 19 April 2010 Jarad Niemi (UCSB) Midterm Review I 19 April 2010 1 / 26
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Class overview Announcements Announcements Homework Homework 3 due today by 4pm in South Hall 5521 Mid-term 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
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Class overview Goals Applied examples Happy Birthday Cakes Predicting UCSB presidential elections Jarad Niemi (UCSB) Midterm Review I 19 April 2010 3 / 26
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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
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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
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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
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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? Bake enough so the probability of being out-of-stock is less than 10%. Minimize expect cost Jarad Niemi (UCSB) Midterm Review I 19 April 2010 7 / 26
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Happy Birthday Cakes Out-of-stock Mathematically, how can we write Bake enough so the probability of being out-of-stock is less than 0.10. 0
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