Let X1, ..., Xn be a random sample from the PDF 2x { 0 < x < 0 0 < x < 1 otherwise, 2(1-a) f(x | 0) = 1-0 where 0 E [0, 1]. Q2.1 Find the method of moments estimator of 0, 0. No files uploaded Q2.2 Is O unbiased? Explain. No files uploaded Q2.3 Find the variance of 0. Is 0 a consistent estimator of 0?
Let X1, ..., Xn be a random sample from the PDF 2x { 0 < x < 0 0 < x < 1 otherwise, 2(1-a) f(x | 0) = 1-0 where 0 E [0, 1]. Q2.1 Find the method of moments estimator of 0, 0. No files uploaded Q2.2 Is O unbiased? Explain. No files uploaded Q2.3 Find the variance of 0. Is 0 a consistent estimator of 0?
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
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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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![**Probability Density Function (PDF) and Estimation Problems**
Let \( X_1, \ldots, X_n \) be a random sample from the PDF
\[
f(x \mid \theta) =
\begin{cases}
\frac{2x}{\theta} & 0 < x < \theta \\
\frac{2(1-x)}{1-\theta} & \theta < x < 1 \\
0 & \text{otherwise,}
\end{cases}
\]
where \( \theta \in [0, 1] \).
**Q2.1**
Find the method of moments estimator of \( \theta \), \( \hat{\theta} \).
**Q2.2**
Is \( \hat{\theta} \) unbiased? Explain.
**Q2.3**
Find the variance of \( \hat{\theta} \). Is \( \hat{\theta} \) a consistent estimator of \( \theta \)?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc58b2eee-5b79-4c9d-923b-d4e913d44dbe%2F1d6f2a1e-f5e9-4a99-92f4-20db50fc134e%2Fgbhfehj_processed.png&w=3840&q=75)
Transcribed Image Text:**Probability Density Function (PDF) and Estimation Problems**
Let \( X_1, \ldots, X_n \) be a random sample from the PDF
\[
f(x \mid \theta) =
\begin{cases}
\frac{2x}{\theta} & 0 < x < \theta \\
\frac{2(1-x)}{1-\theta} & \theta < x < 1 \\
0 & \text{otherwise,}
\end{cases}
\]
where \( \theta \in [0, 1] \).
**Q2.1**
Find the method of moments estimator of \( \theta \), \( \hat{\theta} \).
**Q2.2**
Is \( \hat{\theta} \) unbiased? Explain.
**Q2.3**
Find the variance of \( \hat{\theta} \). Is \( \hat{\theta} \) a consistent estimator of \( \theta \)?
Expert Solution
Step 1
We want to estimate θ, and check it is unbiased or not and also check it is consistent or not .
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