ec41f10_hw3_sol (1)

ec41f10_hw3_sol (1) - Kata Bognar kbognar@ucla.edu...

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Kata Bognar kbognar@ucla.edu Economics 41 Statistics for Economists UCLA Spring 2010 Homework Assignment 3. - suggested solutions by Miao Sun - Bayes Theorem. 1. Dan’s Diner employs three dishwasher. Al washes 40% of the dishes and only breaks 1% of those he handles. Betty and Chuck each wash 30% of the dishes, and Betty breaks only 1% of hers, but Chuck breaks 3% of the dishes he washes. You go to Dan’s for supper one night and hear a dish break at the sink. What is the probability that Chuck is on the job? Answer: Define the following events. A = “Al washes the dishes the night you go for dinner” , B = “Betty washes the dishes the night you go for dinner” , C = “Chuck washes the dishes the night you go for dinner” BR = “a dish breaks the night you go for dinner” . Then the following probabilities are given in the example: P ( A ) = 0 . 4 , P ( B ) = 0 . 3 and P ( C ) = 0 . 3 P ( BR | A ) = 0 . 01 , P ( BR | B ) = 0 . 01 and P ( BR | C ) = 0 . 03 You have to find the conditional probability P ( C | BR ) and the easiest way to do this is to use the Bayes Theorem. P ( C | BR ) = P ( C ) P ( BR | C ) P ( A ) P ( BR | A ) + P ( B ) P ( BR | B ) + P ( C ) P ( BR | C ) = 0 . 3 · 0 . 03 0 . 4 · 0 . 01 + 0 . 3 · 0 . 01 + 0 . 3 · 0 . 03 = 0 . 5625 Discrete Distributions. 2. X is a random variable with a p.m.f f ( x ) = ± 50 x ² 0 . 3 x 0 . 7 50 - x x = 0 , 1 , 2 ,... 50 . Then, C (a) X has binomial distribution with the parameters 50 and 0 . 5 . (b) X has a poisson distribution with a parameter 50 . (c) X has binomial distribution with the parameters 50 and 0 . 3 . (d) X has a poisson distribution with a parameter x. Explanation: The probability mass function of a binomial random variable with parameters n,p is given by f ( x ) = ± n x ² p x (1 - p ) n - x . The probability mass function of a poisson random variable with a parameter λ is given by f ( x ) = e - λ λ x x ! . The p.m.f given above is clearly the p.m.f. of a binomial distribution with parameters n = 50 and p = 0 . 3 . 1
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3. According to an article, 35% of American adults have experienced a breakup at least once during the last 10 years. Of nine randomly selected American adults, find the probability that (a) exactly 5 (b) at most 6 have experienced a breakup at least once during the last 10 years. Let X denote the number of
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This note was uploaded on 12/28/2010 for the course ECON 41 taught by Professor Guggenberger during the Fall '07 term at UCLA.

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ec41f10_hw3_sol (1) - Kata Bognar kbognar@ucla.edu...

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