Math Problem Statement
Kerjakan no 20 dan 21
Solution
From the image you uploaded, there are two problems (No. 20 and No. 21) under "Soal Latihan," which require solving using specific theorems. Here's the breakdown:
Problem 20:
You're given a probability density function (PDF):
\begin{cases} 2\theta^2 x^{-3}, & \text{if } \theta \leq x, \\ 0, & \text{otherwise}. \end{cases}$$ The task is to apply **Theorems 10.4.5 and 10.4.6** to determine a sufficient statistic for $$\theta$$. ### Solution: 1. **Find a sufficient statistic** for $$\theta$$: Using **Theorem 10.4.5** (likely referring to the **Factorization Theorem**), a statistic $$T(X)$$ is sufficient for $$\theta$$ if the likelihood function $$L(\theta; x)$$ can be factored into two parts: $$L(\theta; x) = g(T(x), \theta) \cdot h(x),$$ where $$g(T(x), \theta)$$ depends on the parameter $$\theta$$ and $$T(x)$$, and $$h(x)$$ does not depend on $$\theta$$. 2. **Construct the likelihood function**: For a sample $$x_1, x_2, \dots, x_n$$, the likelihood function is: $$L(\theta; x_1, \dots, x_n) = \prod_{i=1}^{n} f(x_i; \theta) = \prod_{i=1}^{n} 2\theta^2 x_i^{-3}.$$ So, $$L(\theta; x_1, \dots, x_n) = 2^n \theta^{2n} \prod_{i=1}^{n} x_i^{-3}.$$ 3. **Factor the likelihood**: $$L(\theta; x_1, \dots, x_n) = \left( 2^n \prod_{i=1}^{n} x_i^{-3} \right) \cdot \theta^{2n}.$$ Based on the factorization theorem, we can conclude that the statistic $$T(X) = \prod_{i=1}^{n} x_i$$ is sufficient for $$\theta$$, as the second factor depends on $$\theta$$ and the first factor does not. ### Problem 21: This asks you to repeat Problem 20 but with a new distribution: $$f(x; \theta_1, \theta_2) = \begin{cases} \frac{1}{\theta_2 - \theta_1}, & \theta_1 \leq x \leq \theta_2, \\ 0, & \text{otherwise}. \end{cases}$$ Again, use **Theorems 10.4.5 and 10.4.6** to determine a sufficient statistic for $$\theta_1$$ and $$\theta_2$$. ### Solution: 1. **Construct the likelihood function**: For a sample $$x_1, x_2, \dots, x_n$$, the likelihood function is: $$L(\theta_1, \theta_2; x_1, \dots, x_n) = \prod_{i=1}^{n} \frac{1}{\theta_2 - \theta_1}.$$ So, $$L(\theta_1, \theta_2; x_1, \dots, x_n) = \frac{1}{(\theta_2 - \theta_1)^n}.$$ 2. **Factor the likelihood**: The likelihood does not depend directly on the sample values $$x_i$$, but the boundaries of the sample affect the range of $$\theta_1$$ and $$\theta_2$$. The sufficient statistic for this problem is likely related to the minimum and maximum of the sample, since those determine the range of the uniform distribution. Thus, the sufficient statistics for $$\theta_1$$ and $$\theta_2$$ are: $$T(X) = (\min(x_1, \dots, x_n), \max(x_1, \dots, x_n)).$$ --- Let me know if you'd like any further clarification or details about these solutions. ### Related questions: 1. What is the significance of the Factorization Theorem in determining sufficient statistics? 2. How do you apply the concept of likelihood in the case of uniform distributions? 3. Can you explain the difference between sufficient and complete statistics? 4. How do the theorems (10.4.5 and 10.4.6) apply to more complex distributions? 5. What are the implications of finding a sufficient statistic in terms of hypothesis testing? **Tip**: For problems involving uniform distributions, the minimum and maximum values of the sample often serve as sufficient statistics for the parameters defining the distribution.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Probability Density Functions
Sufficient Statistics
Factorization Theorem
Formulas
f(x; θ) = { 2θ^2x^(-3), θ ≤ x; 0 otherwise }
f(x; θ1, θ2) = { 1/(θ2 - θ1), θ1 ≤ x ≤ θ2; 0 otherwise }
L(θ; x1, ..., xn) = Π 2θ^2x_i^(-3)
L(θ1, θ2; x1, ..., xn) = Π 1/(θ2 - θ1)
Theorems
Factorization Theorem
Sufficient Statistic Theorem
Suitable Grade Level
University Level (Statistics/Mathematics)
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