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1. There are at most 100 identical balls in a bowl but we don't know the exact number of them. However, we know that the balls are numbered with numbers 1, 2, ..., 0 and each of the number from number on to number 0 is used exactly once and the 0 ≥ 1 is an integer. We pick a single ball in random. Let Y be the number in a ball we pick at random. (a) What is the parameter space in the statistical model corresponding this setup? (b) What is the probability mass function f(y; 0) of the random variable Y? (c) Suppose that the obseverd number is = y 57. Tell what is the likelihood function corresponding to this data. What is the maximum likelihood estimate of the parameter with this observation. What is the maximum likelihood estimator of the parameter 0? (d) Show by calculating that the espected value of the maximum likelihood estima- tor (Y) is E(Y) = = 0 + 1 2 What is the bias of the ML estimator bias, (⑦(Y))? (Hint: the English Wikipedia page https://en.wikipedia.org/wiki/Discrete_uniform_distribution might be useful)

1. There are at most 100 identical balls in a bowl but we don't know the exact number of them. However, we know that the balls are numbered with numbers 1, 2, ..., 0 and each of the number from number on to number 0 is used exactly once and the 0 ≥ 1 is an integer. We pick a single ball in random. Let Y be the number in a ball we pick at random. (a) What is the parameter space in the statistical model corresponding this setup? (b) What is the probability mass function f(y; 0) of the random variable Y? (c) Suppose that the obseverd number is = y 57. Tell what is the likelihood function corresponding to this data. What is the maximum likelihood estimate of the parameter with this observation. What is the maximum likelihood estimator of the parameter 0? (d) Show by calculating that the espected value of the maximum likelihood estima- tor (Y) is E(Y) = = 0 + 1 2 What is the bias of the ML estimator bias, (⑦(Y))? (Hint: the English Wikipedia page https://en.wikipedia.org/wiki/Discrete_uniform_distribution might be useful)

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.7: Distinguishable Permutations And Combinations
Problem 29E
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1. There are at most 100 identical balls in a bowl but we don't know the exact number of them. However, we know that the balls are numbered with numbers 1, 2, ..., 0 and each of the number from number on to number 0 is used exactly once and the 0 ≥ 1 is an integer. We pick a single ball in random. Let Y be the number in a ball we pick at random. (a) What is the parameter space in the statistical model corresponding this setup? (b) What is the probability mass function f(y; 0) of the random variable Y? (c) Suppose that the obseverd number is = y 57. Tell what is the likelihood function corresponding to this data. What is the maximum likelihood estimate of the parameter with this observation. What is the maximum likelihood estimator of the parameter 0? (d) Show by calculating that the espected value of the maximum likelihood estima- tor (Y) is E(Y) = = 0 + 1 2 What is the bias of the ML estimator bias, (⑦(Y))? (Hint: the English Wikipedia page https://en.wikipedia.org/wiki/Discrete_uniform_distribution might be useful)
Transcribed Image Text:1. There are at most 100 identical balls in a bowl but we don't know the exact number of them. However, we know that the balls are numbered with numbers 1, 2, ..., 0 and each of the number from number on to number 0 is used exactly once and the 0 ≥ 1 is an integer. We pick a single ball in random. Let Y be the number in a ball we pick at random. (a) What is the parameter space in the statistical model corresponding this setup? (b) What is the probability mass function f(y; 0) of the random variable Y? (c) Suppose that the obseverd number is = y 57. Tell what is the likelihood function corresponding to this data. What is the maximum likelihood estimate of the parameter with this observation. What is the maximum likelihood estimator of the parameter 0? (d) Show by calculating that the espected value of the maximum likelihood estima- tor (Y) is E(Y) = = 0 + 1 2 What is the bias of the ML estimator bias, (⑦(Y))? (Hint: the English Wikipedia page https://en.wikipedia.org/wiki/Discrete_uniform_distribution might be useful)
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