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(a) Find the MLE of 0. (b) Find the MLE of Po(X > 1). (c) Prove that the MLE you find in (b) is a consistent estimator of the probability Po(X > 1).

(a) Find the MLE of 0. (b) Find the MLE of Po(X > 1). (c) Prove that the MLE you find in (b) is a consistent estimator of the probability Po(X > 1).

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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**Problem Statement:** Suppose \( X_1, \ldots, X_n \) are independent and identically distributed (iid) with probability density function (pdf) \[ f(x; \theta) = \frac{1}{\theta} e^{-x/\theta}, \quad 0 < x < \infty, \] zero elsewhere. **Tasks:** (a) Find the Maximum Likelihood Estimator (MLE) of \(\theta\). (b) Find the MLE of \( P_\theta(X > 1) \). (c) Prove that the MLE you find in (b) is a consistent estimator of the probability \( P_\theta(X > 1) \).
Transcribed Image Text:**Problem Statement:** Suppose \( X_1, \ldots, X_n \) are independent and identically distributed (iid) with probability density function (pdf) \[ f(x; \theta) = \frac{1}{\theta} e^{-x/\theta}, \quad 0 < x < \infty, \] zero elsewhere. **Tasks:** (a) Find the Maximum Likelihood Estimator (MLE) of \(\theta\). (b) Find the MLE of \( P_\theta(X > 1) \). (c) Prove that the MLE you find in (b) is a consistent estimator of the probability \( P_\theta(X > 1) \).
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