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Thread: cinfidence interval help

  1. #1
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    cinfidence interval help

    Consider independent random samples from two exponential distributions$\displaystyle X_i \sim Exp(\theta_1)$ and $\displaystyle Y_j \sim Exp(\theta_2)$;$\displaystyle i=1,2,...,n_1$ , $\displaystyle j=1,2,....n_2$.
    a) show that$\displaystyle (\theta_2/\theta_1)(\overline{X}/\overline{Y})\sim F(2n_1,2n_2)$.
    b) Derive a 100$\displaystyle r$% confidence interval for $\displaystyle (\theta_2/\theta_1)$.
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  2. #2
    MHF Contributor matheagle's Avatar
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    $\displaystyle X_i/\theta_1\sim EXP(1)=\Gamma(1,1)$, so $\displaystyle \sum_{i=1}^{n_1}X_i/\theta_1\sim\Gamma(n_1,1)$.

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

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