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Suppose $Y_1$,...$Y_n$ is a random sample from a gamma distribution with $\alpha = 2$ and unknown $\beta$.

GOAL: Use $2\sum_{i=1}^n Y_i/\beta$ which is a pivotal quantity to derive a 95% confidence interval for $\beta$

First question, could there be more than one 95% confidence interval satisfies this question?

The one I am given is $$P(\chi_.975^2 \le 2\sum_{i=1}^n Y_i/\beta \le \chi_.025^2) = .95$$

After several intermediate steps which will get us to $$[\frac{2\sum_{i=1}^n Y_i}{\chi_.975^2}, \frac{2\sum_{i=1}^n Y_i}{\chi_.025^2}]$$ is the 95% confidence interval for $\beta$

Is the the right interval as I am not sure $$P(\chi_.975^2 \le 2\sum_{i=1}^n Y_i \le \chi_.025^2) = .95$$ or $$P(\chi_.025^2 \le 2\sum_{i=1}^n Y_i \le \chi_.975^2) = .95$$ or both are the same thing?

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  • $\begingroup$ If you are estimating $\beta$ then it should probably not appear in $\displaystyle \left[\frac{2\sum_{i=1}^n Y_i/\beta}{\chi_.975^2}, \frac{2\sum_{i=1}^n Y_i/\beta}{\chi_.025^2}\right]$ $\endgroup$ – Henry May 2 '14 at 6:12
  • $\begingroup$ Which is larger: $\chi_{.975}^2$ or $\chi_{.025}^2$? $\endgroup$ – Henry May 2 '14 at 6:14
  • $\begingroup$ $\chi_.025^2$ is bigger. Please take a look of my revised confidence interval and also could there be more than one 95% confidence interval satisfies this question? $\endgroup$ – afsdf dfsaf May 2 '14 at 6:34
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I am suprised that you say $\chi_{.975}^2 \lt \chi_{.025}^2$, but if it is true then $\displaystyle P\left(\chi_.025^2 \le 2\sum_{i=1}^n Y_i \le \chi_.975^2\right) = 0$ which is not what you want.

There will be other 95% confidence interval. Examples would include the one-sided cases$$ \displaystyle \left[\frac{2\sum_{i=1}^n Y_i}{\chi_.95^2}, \infty\right) \text{ or } \displaystyle \left(-\infty,\frac{2\sum_{i=1}^n Y_i}{\chi_.05^2}\right]$$

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  • $\begingroup$ If $\chi_.975^2 < \chi_.025^2, \chi_.025^2$ should be in the right hand side like $$P(\chi_.975^2 \le 2\sum_{i=1}^n Y_i/\beta \le \chi_.025^2) = .95$$, right? $\endgroup$ – afsdf dfsaf May 2 '14 at 11:21

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