S231 Lecture 4 Cyntha

# F f y y y 1 n y for y 0 1 n where represents

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Unformatted text preview: inomial random variable with p.f. f (y ; ) = (y ) y (1 )n y for y = 0, 1, ..., n where represents the fraction of the entire population (what population?) that agree. Observed data: y = 520, n = 2000 Likelihood function: L( ) = (2000 ) 520 (1 )2000 520 ^ ) where = y /n (show this!) ^ max L( ) = L( 520 for 0 < < 1 Statistics and Actuarial Science () Model Fitting, Estimation and Checking Jan 2013 17 / 45 Maximum Likelihood Estimation 3 What does "accurate within was it obtained? 2.2%, 19 times out of 20" mean and how Suppose Y represents the number of individuals who, in a randomly selected group of n persons, agreed with the statement. Then Y is n Binomial random variable with p.f. f (y ; ) = (y ) y (1 )n y for y = 0, 1, ..., n where represents the fraction of the entire population (what population?) that agree. Observed data: y = 520, n = 2000 Likelihood function: L( ) = (2000 ) 520 (1 )2000 520 ^ ) where = y /n (show this!) ^ max L( ) = L( 520 for 0 < < 1 ^ Maximum likelihood estimate of is = 520/2000 or 26% Statistics and Actuarial Science () Model Fitting, Estimation and Checking Jan 2013 17 / 45 Likelihood Function for Harris Decima Poll From the graph, the interval suggested by the pollsters, 26% 2.2% or [23.8, 28.2] is reasonable since it seems to contain most of the values of with large values of the likelihood L( ). (More on interval estimates in Chapter 4). Statistics and Actuarial Science () Model Fitting, Estimation and Checking Jan 2013 18 / 45 Graph of L() for n=2000 and y=520 (L() is often rescaled so max=1) Graph of L() for n=200 and y=52 Is this the same picture? Its maximum is in the same place. Graph of L() for n=2000 and n=200 HIV example. Suppose Y represents the number of persons infected with the human immunode...ciency virus (HIV) in a randomly selected group of n persons. Modelled by Binomial P (Y = y ; ) = f (y ; ) = n y (1 y )n y for y = 0, 1, . . . , n where is the fraction of the population that are infected. In a random sample of n persons tested for HIV, y = y is the observed number...
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## This note was uploaded on 04/17/2013 for the course STAT 231 taught by Professor Cantremember during the Winter '08 term at Waterloo.

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