# Probability that one random variable is greater than other

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.

• 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. – BGM Feb 7 at 18:31
• @BGM, thank you for pointing that out. Updated the question. – jhon_wick Feb 7 at 19:47
• @BGM, do you think the numerator is right: $\int_\gamma^\infty \int^{\infty}_{x_1} f_{x_1,x_2} dx_2 dx_1$? – jhon_wick Feb 7 at 20:51
• It looks good to me. – BGM Feb 8 at 3:12