I am working on confidence intervals and would like to know if I did it correctly.
Let $X$ be a random variable. Find the $100(1-\alpha)$% CI for the parameter $\theta$ when
1), $X$ is Exponential with mean $\theta$.
2), $X$ is Uniform$(0,\theta)$.
3), $X_1, X_2, ..., X_n$ are iid Uniform$(0,\theta)$.
I simply used the inequality
$$a \le U \le b$$ $$Pr[U \le a]=\frac{\alpha}{2}$$ $$Pr[b \le U=\frac{\alpha}{2}$$
where $U=\frac{X}{\theta}$ is the pivot which resulted in
1), $\frac{X}{-\ln{(\alpha/2)}} \le \theta \le \frac{X}{-\ln{(1-\alpha/2)}}$
2), $\frac{X}{\alpha/2} \le \theta \le \frac{X}{1-\alpha/2}$
3), $\frac{X}{\sqrt[n]{\alpha/2}} \le \theta \le \frac{X}{\sqrt[m]{1-\alpha/2}}$
I have an exam in 2 days, it would be great if you could confirm or let me know if I made some mistake.
Thanks!