cinfidence interval help

**Kat-M** · May 13th 2009, 12:18 AM

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 $(\theta_2/\theta_1)$ .

**matheagle** · May 14th 2009, 11:28 PM

$X_i/\theta_1\sim EXP(1)=\Gamma(1,1)$ , 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.

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