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1. Suppose that Y; takes values 0, 1, 2 with probability f(y;0) = e®y-b(@). (a) Find the function b(0) so that f(y; 0) is a probability mass function (i.e., total probability one). Find the values f(0; 0), f(1;0), and f(2;0) for 0 = 0, 0 = log .5, 0 = log 2 (remember that unless otherwise specified, all logs are natural logs). Find the expected value of Y for each of these three values of 0. (b) If we observe Y1,Y2, ..., Yn, find the log-likelihood function ((0 ;Y1,Y2, ...,Y,). (c) Suppose that we have a sample of size 50 and observe the following summary of values of Y;: Y = 0 Y =1 Y = 2 Total 26 11 13 50 (I.e., 11 of the Y; have value 0, etc.) Find the mean of the Y; and compare to the means from part (a). What does this tell you about the MLE for 0?

1. Suppose that Y; takes values 0, 1, 2 with probability f(y;0) = e®y-b(@). (a) Find the function b(0) so that f(y; 0) is a probability mass function (i.e., total probability one). Find the values f(0; 0), f(1;0), and f(2;0) for 0 = 0, 0 = log .5, 0 = log 2 (remember that unless otherwise specified, all logs are natural logs). Find the expected value of Y for each of these three values of 0. (b) If we observe Y1,Y2, ..., Yn, find the log-likelihood function ((0 ;Y1,Y2, ...,Y,). (c) Suppose that we have a sample of size 50 and observe the following summary of values of Y;: Y = 0 Y =1 Y = 2 Total 26 11 13 50 (I.e., 11 of the Y; have value 0, etc.) Find the mean of the Y; and compare to the means from part (a). What does this tell you about the MLE for 0?

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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1. Suppose that Y; takes values 0, 1, 2 with probability f (y;0) = e®y-b(0). (a) Find the function b(0) so that f(y; 0) is a probability mass function (i.e., total probability one). Find the values f(0; 0), f(1;0), and f(2;0) for 0 = 0, 0 = log .5, 0 = log 2 (remember that unless otherwise specified, all logs are natural logs). Find the expected value of Y for each of these three values of 0. (b) If we observe Y1,Y2, ..., Yn, find the log-likelihood function l(0 ;Y1,Y2, ..., Y,). (c) Suppose that we have a sample of size 50 and observe the following summary of values of Y;: Y = 0 Y = 1 Y = 2 Total 11 13 26 50 (L.e., 11 of the Y; have value 0, etc.) Find the mean of the Y; and compare to the means from part (a). What does this tell you about the MLE for 0? (d) Graph twice the log-likelihood function against 0 given these data for 0 over the interval (0,1). From the graph estimate the MLE for 0 and an approximate 95% confidence interval. (e) Find the score function U(0) (suppressing dependence on Y1,..., Yn) and evaluate at 0 = 0. (f) Find the Fisher information, I(0) and evaluate at 0 = 0. (g) Conduct the score test of the null hypothesis Ho: 0 = 0. If µ is the expected value of Y when 0 = 0 (from part (a)), compare to the one sample t-test of Ho: E[Y] = µ.
Transcribed Image Text:1. Suppose that Y; takes values 0, 1, 2 with probability f (y;0) = e®y-b(0). (a) Find the function b(0) so that f(y; 0) is a probability mass function (i.e., total probability one). Find the values f(0; 0), f(1;0), and f(2;0) for 0 = 0, 0 = log .5, 0 = log 2 (remember that unless otherwise specified, all logs are natural logs). Find the expected value of Y for each of these three values of 0. (b) If we observe Y1,Y2, ..., Yn, find the log-likelihood function l(0 ;Y1,Y2, ..., Y,). (c) Suppose that we have a sample of size 50 and observe the following summary of values of Y;: Y = 0 Y = 1 Y = 2 Total 11 13 26 50 (L.e., 11 of the Y; have value 0, etc.) Find the mean of the Y; and compare to the means from part (a). What does this tell you about the MLE for 0? (d) Graph twice the log-likelihood function against 0 given these data for 0 over the interval (0,1). From the graph estimate the MLE for 0 and an approximate 95% confidence interval. (e) Find the score function U(0) (suppressing dependence on Y1,..., Yn) and evaluate at 0 = 0. (f) Find the Fisher information, I(0) and evaluate at 0 = 0. (g) Conduct the score test of the null hypothesis Ho: 0 = 0. If µ is the expected value of Y when 0 = 0 (from part (a)), compare to the one sample t-test of Ho: E[Y] = µ.
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