I have two independent Rayleigh distributed random variable: $X_1$ and $X_2$. I would like to derive the probability of $P(X_1 < X_2| X_1 > \gamma)$.

I started the derivation as: $P(X_1 < X_2| X_1 > \gamma) = \frac{\int_\gamma^\infty \left[ \int^{\infty}_{x_1} f_{x_1,x_2} dx_2 \right] dx_1}{P(X_1 > \gamma)}$

Then, $\frac{\int_\gamma^\infty \left[ \int^{\infty}_{x_1} f_{x_1,x_2} dx_2 \right] dx_1}{1 - \int_{-\infty}^\gamma f_{x_1}dx_1}$.

Am I doing it correctly? Thank you.

  • 1
    $\begingroup$ Where did the $y$ pops out? The numerator is $\Pr\{X_1 > \gamma, X_1 < X_2\}$, and the integration limit follows accordingly, and depends on the order of integration you choose. $\endgroup$ – BGM Feb 7 at 18:31
  • $\begingroup$ @BGM, thank you for pointing that out. Updated the question. $\endgroup$ – jhon_wick Feb 7 at 19:47
  • $\begingroup$ @BGM, do you think the numerator is right: $\int_\gamma^\infty \int^{\infty}_{x_1} f_{x_1,x_2} dx_2 dx_1$? $\endgroup$ – jhon_wick Feb 7 at 20:51
  • 1
    $\begingroup$ It looks good to me. $\endgroup$ – BGM Feb 8 at 3:12

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