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Math Help - cinfidence interval help

  1. #1
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    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).
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  2. #2
    MHF Contributor matheagle's Avatar
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    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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