Math Problem Statement

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Let X1, X2, ..., Xn be a random sample of size n from a distribution with a probability density function f(x; θ) = θx^(θ-1) if 0 < x < 1, 0 otherwise, where 0 < θ is a parameter. Using the maximum likelihood method, find an estimator for the parameter θ.

Solution

To solve this problem, we aim to find the maximum likelihood estimator (MLE) for the parameter θ\theta based on the given probability density function (pdf). Here's the step-by-step explanation:

1. Probability Density Function (PDF)

The given pdf is:

\begin{cases} \theta x^{\theta - 1}, & \text{if } 0 < x < 1, \\ 0, & \text{otherwise}. \end{cases}$$ where $$\theta > 0$$ is the parameter to estimate. ### 2. Likelihood Function Given a random sample $$X_1, X_2, \ldots, X_n$$ from this distribution, the joint pdf (likelihood function) is the product of individual pdfs: $$L(\theta) = \prod_{i=1}^n f(X_i; \theta),$$ where $$f(X_i; \theta) = \theta X_i^{\theta - 1}$$ for $$0 < X_i < 1$$. Substituting $$f(X_i; \theta)$$ into $$L(\theta)$$, we get: $$L(\theta) = \prod_{i=1}^n \theta X_i^{\theta - 1} = \theta^n \prod_{i=1}^n X_i^{\theta - 1}.$$ ### 3. Log-Likelihood Function Taking the natural logarithm of the likelihood function (to simplify the calculations): $$\ell(\theta) = \ln L(\theta) = \ln(\theta^n) + \ln\left(\prod_{i=1}^n X_i^{\theta - 1}\right).$$ Using logarithm rules: $$\ell(\theta) = n \ln(\theta) + (\theta - 1) \sum_{i=1}^n \ln(X_i).$$ ### 4. Derivative of Log-Likelihood To find the maximum likelihood estimator, we differentiate $$\ell(\theta)$$ with respect to $$\theta$$: $$\frac{d\ell(\theta)}{d\theta} = \frac{n}{\theta} + \sum_{i=1}^n \ln(X_i).$$ Set $$\frac{d\ell(\theta)}{d\theta} = 0$$ to find the critical points: $$\frac{n}{\theta} + \sum_{i=1}^n \ln(X_i) = 0.$$ Rearranging: $$\frac{n}{\theta} = -\sum_{i=1}^n \ln(X_i).$$ Solving for $$\theta$$: $$\theta = -\frac{n}{\sum_{i=1}^n \ln(X_i)}.$$ ### 5. Maximum Likelihood Estimator The MLE for $$\theta$$ is: $$\hat{\theta} = -\frac{n}{\sum_{i=1}^n \ln(X_i)}.$$ ### Explanation - $$\sum_{i=1}^n \ln(X_i)$$ represents the sum of the natural logarithms of the sample values. - The negative sign arises because $$\ln(X_i)$$ for $$0 < X_i < 1$$ is negative, making $$\hat{\theta}$$ positive, as required ($$\theta > 0$$). Let me know if you need clarification or further details on any step! --- ### Questions for Deeper Understanding 1. What is the intuition behind the likelihood function in statistical inference? 2. How does taking the logarithm simplify the computation of the MLE? 3. Why do we require $$\theta > 0$$ in the context of this problem? 4. How does the sample size $$n$$ affect the precision of the MLE for $$\theta$$? 5. Can this MLE be biased, and if so, how would you calculate its bias? ### Tip When working with likelihood functions, always confirm that the resulting critical point maximizes the likelihood by checking the second derivative or the shape of the

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Math Problem Analysis

Mathematical Concepts

Maximum Likelihood Estimation (MLE)
Probability Density Functions
Parameter Estimation

Formulas

f(x; θ) = θx^(θ-1), if 0 < x < 1
Log-likelihood: l(θ) = n * ln(θ) + (θ - 1) * ∑ln(Xi)
MLE: θ = -n / ∑ln(Xi)

Theorems

Maximum Likelihood Estimation

Suitable Grade Level

Undergraduate (Statistics or Probability Course)

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