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In lecture (Mon 1/30), we showed that for Bernoulli (p) distribution, the MLE estimator p = x Exercise 1 is sufficient for the parameter p; for Uniform([a, b]), the MLE estimators Geometric (p) are jointly sufficient for the parameters a, b. In this exercise, you will deduce similar results for the following four distributions: • Exp(A) • Poisson(X) • N(μ1,0²) â = y₁ = min(₁, ...,xn), b = y₁ := max(x1, ***, En) (i) For each of these four distributions, write down their likelihood functions. (Hint: the log-likelihood functions for these distributions were computed in previous lecture and homework.) (ii) Use the Fisher-Neyman factorization theorem to show that is a sufficient statistic for each of the first three distributions. (iii) Show that 1/2 is also a sufficient statistic for each of these three distributions. (iv) Use the Fisher-Neyman factorization theorem to show that , v are a pair of jointly sufficient statistics for N(μ, 0²).

In lecture (Mon 1/30), we showed that for Bernoulli (p) distribution, the MLE estimator p = x Exercise 1 is sufficient for the parameter p; for Uniform([a, b]), the MLE estimators Geometric (p) are jointly sufficient for the parameters a, b. In this exercise, you will deduce similar results for the following four distributions: • Exp(A) • Poisson(X) • N(μ1,0²) â = y₁ = min(₁, ...,xn), b = y₁ := max(x1, ***, En) (i) For each of these four distributions, write down their likelihood functions. (Hint: the log-likelihood functions for these distributions were computed in previous lecture and homework.) (ii) Use the Fisher-Neyman factorization theorem to show that is a sufficient statistic for each of the first three distributions. (iii) Show that 1/2 is also a sufficient statistic for each of these three distributions. (iv) Use the Fisher-Neyman factorization theorem to show that , v are a pair of jointly sufficient statistics for N(μ, 0²).

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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part iv (i already asked other parts)

Exercise 1 In lecture (Mon 1/30), we showed that for Bernoulli (p) distribution, the MLE estimator p=i is sufficient for the parameter p; • for Uniform([a, b]), the MLE estimators • Poisson(X) • N(μ1,0²) â = y₁ = min(x₁, · · , £n), b = Yn = 1 max(T₁,..., In) are jointly sufficient for the parameters a, b. In this exercise, you will deduce similar results for the following four distributions: • Exp(X) • Geometric (p) (i) For each of these four distributions, write down their likelihood functions. (Hint: the log-likelihood functions for these distributions were computed in previous lecture and homework.) (ii) Use the Fisher-Neyman factorization theorem to show that is a sufficient statistic for each of the first three distributions. (iii) Show that 1/7 is also a sufficient statistic for each of these three distributions. (iv) Use the Fisher-Neyman factorization theorem to show that I, v are a pair of jointly sufficient statistics for N(μ, 0²).
Transcribed Image Text:Exercise 1 In lecture (Mon 1/30), we showed that for Bernoulli (p) distribution, the MLE estimator p=i is sufficient for the parameter p; • for Uniform([a, b]), the MLE estimators • Poisson(X) • N(μ1,0²) â = y₁ = min(x₁, · · , £n), b = Yn = 1 max(T₁,..., In) are jointly sufficient for the parameters a, b. In this exercise, you will deduce similar results for the following four distributions: • Exp(X) • Geometric (p) (i) For each of these four distributions, write down their likelihood functions. (Hint: the log-likelihood functions for these distributions were computed in previous lecture and homework.) (ii) Use the Fisher-Neyman factorization theorem to show that is a sufficient statistic for each of the first three distributions. (iii) Show that 1/7 is also a sufficient statistic for each of these three distributions. (iv) Use the Fisher-Neyman factorization theorem to show that I, v are a pair of jointly sufficient statistics for N(μ, 0²).
Expert Solution
Step 1

The pdf of Expλ is fx=λ·e-λx for x>0.

The pmf of Geometricp is fx=1-px·p for x=0, 1, 2, ...,∞.

The pmf of Poissonλ is fx=e-λ·λxx! for x=0, 1, 2, ...,∞.

The pdf of Nμ, σ2 is fx=1σ2πe-x-μ22σ2.

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