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Assume that $X_1,X_2$ are independent random variables with exponential distribution with the same mean 100. Let $X_{(1)}=\min\{X_1,X_2\}$ and $X_{(2)}=\max\{X_1,X_2\}$. Calculate $P(e^{-0.01X_{(1)}}+e^{-0.01X_{(2)}}>50)$.

I will be able to solve this exercise if $X_{(1)},X_{(2)}$ would be given by simple function of $X_1,X_2$. However I cannot give $f$ for $(X_{(1)},X_{(2)})=f(X_1,X_2)$. Could you please help to solve the exercise?

Almost sure
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    $e^{-0.01X_{(1)}}+e^{-0.01X_{(2)}}=e^{-0.01X_{1}}+e^{-0.01X_{2}}$ – Augustin May 05 '16 at 12:00
  • Yes, you're right :D so the exercise is solved. But if it would be $e^{−0.01X_{(1)}}-e^{−0.01X_{(2)}}$ instead of $e^{−0.01X_{(1)}}+e^{−0.01X_{(2)}}$? – Almost sure May 05 '16 at 12:04
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    Then you could first try to find the joint distribution of $(X_{(1)},X_{(2)})$. – Augustin May 05 '16 at 12:12
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    have also a look at http://math.stackexchange.com/questions/80475/order-statistics-of-i-i-d-exponentially-distributed-sample – Jean Marie May 05 '16 at 12:13

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