Math Problem Statement
Let X1,· · · , Xn be a random sample from a population with density function
f (x) =
1
a
for 0 ≤ x < a, a > 0
0 otherwise
¯
Show that the moment estimator of a is given by ˆ a = 2
X.
Solution
We are given that is a random sample from a population with a probability density function:
\begin{cases} \frac{1}{a} & \text{for } 0 \leq x < a \\ 0 & \text{otherwise} \end{cases}$$ where $$a > 0$$ is an unknown parameter. Our goal is to find the moment estimator of $$a$$. ### Step 1: Find the first moment (mean) of the distribution The moment estimator is based on matching theoretical moments (such as the mean) to the sample moments. The mean (first moment) of a random variable $$X$$ with the given density function is: $$\mathbb{E}(X) = \int_0^a x f(x) \, dx$$ Substituting the value of $$f(x) = \frac{1}{a}$$ into the equation: $$\mathbb{E}(X) = \int_0^a x \cdot \frac{1}{a} \, dx$$ Now, compute the integral: $$\mathbb{E}(X) = \frac{1}{a} \int_0^a x \, dx = \frac{1}{a} \left[ \frac{x^2}{2} \right]_0^a = \frac{1}{a} \cdot \frac{a^2}{2} = \frac{a}{2}$$ Thus, the expected value of $$X$$ is: $$\mathbb{E}(X) = \frac{a}{2}$$ ### Step 2: Equating the sample mean to the population mean The moment estimator $$\hat{a}$$ is obtained by equating the sample mean $$\overline{X}$$ to the theoretical mean $$\mathbb{E}(X)$$. The sample mean $$\overline{X}$$ is: $$\overline{X} = \frac{1}{n} \sum_{i=1}^n X_i$$ Equating this to the theoretical mean $$\mathbb{E}(X) = \frac{a}{2}$$: $$\overline{X} = \frac{a}{2}$$ Solving for $$a$$, we get: $$a = 2 \overline{X}$$ ### Step 3: Conclusion The moment estimator of $$a$$ is: $$\hat{a} = 2 \overline{X}$$ This is the required moment estimator. --- Would you like more details or have any questions? Here are 5 related questions: 1. How do we calculate higher-order moments for this distribution? 2. What is the second moment of this distribution? 3. How does the maximum likelihood estimator (MLE) of $$a$$ compare to the moment estimator? 4. Why is the moment estimator of $$a$$ proportional to the sample mean? 5. Can we compute the variance of this moment estimator? **Tip**: Moment estimators are not always the most efficient estimators, but they provide a quick way to estimate parameters based on known theoretical moments.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Moment Estimation
Probability Density Functions
Expected Value
Sample Mean
Formulas
f(x) = 1/a for 0 ≤ x < a
E(X) = (1/a) ∫_0^a x dx = a/2
Sample Mean X̄ = (1/n) ∑ X_i
Moment Estimator â = 2X̄
Theorems
Moment Estimation Theorem
Expected Value Theorem for Uniform Distributions
Suitable Grade Level
Undergraduate Statistics
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