Let X₁, X5 be a random sample from a continuous probability distribution with pdf f (x | 0) = if 0
Let X₁, X5 be a random sample from a continuous probability distribution with pdf f (x | 0) = if 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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![## Problem Statement
Let \( X_1, \ldots, X_5 \) be a random sample from a continuous probability distribution with the probability density function (pdf):
\[ f(x \mid \theta) = \frac{1}{\theta} \quad \text{if } 0 < x < \theta \quad \text{and } 0 \text{ otherwise.} \]
### Questions
a. Find the method of moment (MOM) estimator of \( \theta \). Show your workings.
b. A random sample yields data \( x_1 = 0.2 \), \( x_2 = 1.1 \), \( x_3 = 1.7 \), \( x_4 = 0.6 \), \( x_5 = 1.9 \). Compute the estimate for this data.
### Explanation of the Steps for MOM Estimator
#### a) Finding the Method of Moment (MOM) Estimator of \( \theta \)
1. **Identify the sample moments:**
- The first sample moment is the sample mean, \(\bar{X}\).
2. **Determine the population moments:**
- For the given pdf, the mean \( \mu \) can be found as follows:
\[
\mu = E[X] = \int_{0}^{\theta} x f(x \mid \theta) \, dx \\
= \int_{0}^{\theta} x \left(\frac{1}{\theta}\right) \, dx \\
= \frac{1}{\theta} \int_{0}^{\theta} x \, dx \\
= \frac{1}{\theta} \left[\frac{x^2}{2}\right]_0^{\theta} \\
= \frac{1}{\theta} \left(\frac{\theta^2}{2}\right) \\
= \frac{\theta}{2}
\]
3. **Equate the sample moment to the population moment:**
- The MOM estimator \(\hat{\theta}\) is determined by solving the equation:
\[
\bar{X} = \frac{\theta}{2}
\]
Therefore,
\[
\hat{\theta} = 2\bar{X}
\]
#### b) Calculation with Sample Data
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Transcribed Image Text:## Problem Statement
Let \( X_1, \ldots, X_5 \) be a random sample from a continuous probability distribution with the probability density function (pdf):
\[ f(x \mid \theta) = \frac{1}{\theta} \quad \text{if } 0 < x < \theta \quad \text{and } 0 \text{ otherwise.} \]
### Questions
a. Find the method of moment (MOM) estimator of \( \theta \). Show your workings.
b. A random sample yields data \( x_1 = 0.2 \), \( x_2 = 1.1 \), \( x_3 = 1.7 \), \( x_4 = 0.6 \), \( x_5 = 1.9 \). Compute the estimate for this data.
### Explanation of the Steps for MOM Estimator
#### a) Finding the Method of Moment (MOM) Estimator of \( \theta \)
1. **Identify the sample moments:**
- The first sample moment is the sample mean, \(\bar{X}\).
2. **Determine the population moments:**
- For the given pdf, the mean \( \mu \) can be found as follows:
\[
\mu = E[X] = \int_{0}^{\theta} x f(x \mid \theta) \, dx \\
= \int_{0}^{\theta} x \left(\frac{1}{\theta}\right) \, dx \\
= \frac{1}{\theta} \int_{0}^{\theta} x \, dx \\
= \frac{1}{\theta} \left[\frac{x^2}{2}\right]_0^{\theta} \\
= \frac{1}{\theta} \left(\frac{\theta^2}{2}\right) \\
= \frac{\theta}{2}
\]
3. **Equate the sample moment to the population moment:**
- The MOM estimator \(\hat{\theta}\) is determined by solving the equation:
\[
\bar{X} = \frac{\theta}{2}
\]
Therefore,
\[
\hat{\theta} = 2\bar{X}
\]
#### b) Calculation with Sample Data
1.
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