Let X₁, X5 be a random sample from a continuous probability distribution with pdf f (x | 0) = if 0

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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

1.
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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