Math Problem Statement
Let X1, X2, ..., Xn, n >= 5, be a random sample from a distribution with the probability density function f(x; θ) = e^{-(x - θ)} for x >= θ, 0 otherwise, where θ ∈ R is the unknown parameter. Determine which of the following statements about the 95% confidence interval for θ are true:
(a) A 95% confidence interval of θ has to be of finite length.
(b) (min(X1, ..., Xn) + (1/n) ln(0.05), min(X1, ..., Xn)) is a 95% confidence interval for θ.
(c) A 95% confidence interval of θ can be of length 1.
(d) A 95% confidence interval of θ can be of length 2.
Solution
Let's break down the problem you're presenting.
You have a random sample drawn from a distribution with the following probability density function (PDF):
\begin{cases} e^{-(x - \theta)}, & x \geq \theta \\ 0, & \text{otherwise} \end{cases}$$ where $$\theta$$ is the unknown parameter, and the sample size is $$n \geq 5$$. The goal is to determine which of the following statements are true regarding a 95% confidence interval for $$\theta$$: (a) A 95% confidence interval of $$\theta$$ has to be of finite length. (b) $$\left( \min(X_1, X_2, \dots, X_n) + \frac{1}{n} \ln(0.05), \min(X_1, X_2, \dots, X_n) \right)$$ is a 95% confidence interval for $$\theta$$. (c) A 95% confidence interval of $$\theta$$ can be of length 1. (d) A 95% confidence interval of $$\theta$$ can be of length 2. Let's analyze each statement: ### Statement (a) Since $$X_i$$ are from a distribution where the PDF involves $$e^{-(x - \theta)}$$, and for any sample point $$X_i \geq \theta$$, the interval for $$\theta$$ should be finite. Otherwise, the range of possible values for $$\theta$$ would be unbounded. Thus, **statement (a) is true**. ### Statement (b) The form of this confidence interval looks like it's using a transformation of the sample minimum. The confidence interval uses the minimum of the observed sample and adjusts it with a logarithmic term to create bounds. This is consistent with the method of creating confidence intervals for the parameter $$\theta$$ using order statistics (like the minimum in this case). Thus, **statement (b) is also true**. ### Statement (c) It would not be typical for a confidence interval to have a length of exactly 1 since this depends on the data. However, it's not impossible if the data is very close in value. In theory, the length could be 1 depending on the realization of the sample. Thus, **statement (c) can be true** but is less likely in practice. It depends on the data. ### Statement (d) Similarly to (c), the length of the confidence interval could potentially be 2 if the data were to support that interval size. Thus, **statement (d) can be true** as well. ### Conclusion: - Statements (a), (b), (c), and (d) are all plausible, but based on typical practice, (a) and (b) are certainly true. - (c) and (d) are theoretically possible, depending on the sample. Let me know if you need further clarification or details about any of the steps! Would you like to explore confidence intervals for different distributions or other related topics? ### Further Questions: 1. How do confidence intervals work for other distributions, like normal or binomial? 2. Can you describe how the sample size impacts the length of a confidence interval? 3. What are other methods for estimating confidence intervals for unknown parameters? 4. How do you derive the maximum likelihood estimator (MLE) for a given distribution? 5. How would you interpret a confidence interval in a real-world scenario? **Tip:** Always check the assumptions behind any statistical method (e.g., sample size, type of distribution) before applying it!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Confidence intervals
Exponential distribution
Order statistics
Formulas
f(x; θ) = e^{-(x - θ)}, x >= θ
Confidence interval: (min(X1, ..., Xn) + (1/n) ln(α), min(X1, ..., Xn))
Theorems
Likelihood-based confidence intervals
Properties of exponential distributions
Suitable Grade Level
Undergraduate (Statistics/Probability)
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