# Math Help - cinfidence interval help

1. ## cinfidence interval help

Consider independent random samples from two exponential distributions $X_i \sim Exp(\theta_1)$ and $Y_j \sim Exp(\theta_2)$; $i=1,2,...,n_1$ , $j=1,2,....n_2$.
a) show that $(\theta_2/\theta_1)(\overline{X}/\overline{Y})\sim F(2n_1,2n_2)$.
b) Derive a 100 $r$% confidence interval for $(\theta_2/\theta_1)$.

2. $X_i/\theta_1\sim EXP(1)=\Gamma(1,1)$, so $\sum_{i=1}^{n_1}X_i/\theta_1\sim\Gamma(n_1,1)$.

Likewise $\sum_{j=1}^{n_2}Y_j/\theta_2\sim\Gamma(n_2,1)$.

Next, you need to tranform these to be $\chi^2$ random variables.
Then taking the ratio of independent $\chi^2$'s divided by their degrees of freedom should complete this problem.