Let X₁, X2,..., X, * fx(x; 0) be an iid sample from a distribution with pdf [0/xº+¹, if x ≥ 1 0, otherwise for all > 1. 1. fx(x; 0) = Write down the support Sx and the parameter space Ⓒ. Is fx(x; 0) a well-defined pdf? 2. Compute E(X) where X ~ fx(x; 0). Using your answer for E(X) find the method of moments estimator MM in terms of X. 3. Write down the log likelihood function for the samples X₁… X₂, and use this to derive ÎMLE, the maximum likelihood estimator for 0.
Let X₁, X2,..., X, * fx(x; 0) be an iid sample from a distribution with pdf [0/xº+¹, if x ≥ 1 0, otherwise for all > 1. 1. fx(x; 0) = Write down the support Sx and the parameter space Ⓒ. Is fx(x; 0) a well-defined pdf? 2. Compute E(X) where X ~ fx(x; 0). Using your answer for E(X) find the method of moments estimator MM in terms of X. 3. Write down the log likelihood function for the samples X₁… X₂, and use this to derive ÎMLE, the maximum likelihood estimator for 0.
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![**Transcription for Educational Website:**
---
**Problem Statement:**
Let \( X_1, X_2, \ldots, X_n \) be an iid sample from a distribution with pdf
\[
f_X(x; \theta) =
\begin{cases}
\frac{\theta}{x^{\theta + 1}}, & \text{if } x \geq 1 \\
0, & \text{otherwise}
\end{cases}
\]
for all \( \theta > 1 \).
**Questions:**
1. **[Task 1]**
Write down the support \( S_X \) and the parameter space \( \Theta \). Is \( f_X(x; \theta) \) a well-defined pdf?
2. **[Task 2]**
Compute \( \mathbb{E}(X) \) where \( X \sim f_X(x; \theta) \). Using your answer for \( \mathbb{E}(X) \), find the method of moments estimator \( \hat{\theta}_{MM} \) in terms of \( \overline{X} \).
3. **[Task 3]**
Write down the log likelihood function for the samples \( X_1, \ldots, X_n \), and use this to derive \( \hat{\theta}_{MLE} \), the maximum likelihood estimator for \( \theta \).
---
**Note:** No graphs or diagrams are included in this image.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd7c2748e-5ba1-4f2f-bc7d-c6e1ab8bba12%2F47cd900c-e537-4a10-b4e1-d36e362c1bf3%2Fr8cuzi_processed.png&w=3840&q=75)
Transcribed Image Text:**Transcription for Educational Website:**
---
**Problem Statement:**
Let \( X_1, X_2, \ldots, X_n \) be an iid sample from a distribution with pdf
\[
f_X(x; \theta) =
\begin{cases}
\frac{\theta}{x^{\theta + 1}}, & \text{if } x \geq 1 \\
0, & \text{otherwise}
\end{cases}
\]
for all \( \theta > 1 \).
**Questions:**
1. **[Task 1]**
Write down the support \( S_X \) and the parameter space \( \Theta \). Is \( f_X(x; \theta) \) a well-defined pdf?
2. **[Task 2]**
Compute \( \mathbb{E}(X) \) where \( X \sim f_X(x; \theta) \). Using your answer for \( \mathbb{E}(X) \), find the method of moments estimator \( \hat{\theta}_{MM} \) in terms of \( \overline{X} \).
3. **[Task 3]**
Write down the log likelihood function for the samples \( X_1, \ldots, X_n \), and use this to derive \( \hat{\theta}_{MLE} \), the maximum likelihood estimator for \( \theta \).
---
**Note:** No graphs or diagrams are included in this image.
Transcribed Image Text:Let \( Z_1, Z_2, Z_3, Z_4, Z_5, Z_6 \sim \text{iid } N(0, 1) \),
i.e., each \( Z_i \) is iid a standard Normal random variable. Use them to construct
1. A Chi-squared random variable \( \chi^2_3 \) with 3 degrees of freedom.
2. A Student's \( t \) random variable \( t_5 \) with 5 degrees of freedom.
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