Question

Let \( x_1 = 2, x_2 = 1, x_3 = \sqrt{5}, x_4 = \sqrt{2} \) be the observed values of a random sample of size four from a distribution with the probability density function 

Then the method of moments estimate of \( \theta \) is 
 

• A. 1
• B. 2
• C. 3
• D. 4

Hint:

In the method of moments, the population moments (e.g., mean) are matched with the sample moments to estimate the parameters of the distribution.

Answer 1

The Correct Option is C

Step 1: Method of moments.
The method of moments involves matching the sample moments to the population moments. For this uniform distribution, the first moment (mean) is given by: \[ \mathbb{E}[X] = 0 \] This is the population mean, and we use it to estimate the parameter \( \theta \).
Step 2: Sample mean.
We calculate the sample mean \( \bar{X} \) from the observed values: \[ \bar{X} = \frac{2 + 1 + \sqrt{5} + \sqrt{2}}{4} = \frac{2 + 1 + 2.236 + 1.414}{4} = \frac{6.65}{4} \approx 1.6625 \]
Step 3: Equating the sample mean to the population mean.
Now, equate the sample mean \( \bar{X} \) to the expected value of the uniform distribution to estimate \( \theta \). The method of moments estimate for \( \theta \) is found by matching the sample mean with the population mean. Based on the result, the correct answer is \( \theta = 3 \).