# Questions & Answers

## Confidence Intervals

- Let \( \displaystyle X_{1} , \dots X_{n} \) be a random sample from a population with distribution \( \displaystyle Exp\left(\frac{1}{\theta}\right) \), where \( \theta > 0 \). Find a confidence interval for \( \theta \).
- Let \( \displaystyle X_{1} , \dots X_{n} \) be a random sample from a population with density

\[ \displaystyle f(x| \theta_{1},\theta_{2}) = \frac{\theta_{1}}{\theta_{2}} x^{\theta_{2}-1}{e}^{-\frac{x^{\theta_{2}}}{\theta_{1}}} \]

where \( \theta_{1}>0,\theta_{2}>0,x>0 \) and \( \theta_{2} \) is known. Find a confidence interval for \( \theta_{1} \).

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