Ratio Distribution - Question about the solution on wolfram

Question

I just read the calculation of ratio distribution on Wolfram.
I found I cannot understand the following step (From the step 2 to step 3):
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$$ P(Y<=uX|X>0)+P(Y>=ux|X<0)=\int_0^{\infty}\int_0^{ux}f(x,y)dydx+\int_{-\infty}^0\int_{ux}^0f(x,y)dydx$$ Could anyone explain it for me? I think it is a conditional probability, so at least we need to divide it by P(X > 0) or P(X < 0).
Thank you very much!

Answer 1

The probability being conditioned on cancels out for each piece.

\begin{align} &P\left( \frac{Y}X \leq u \right) \\ &= P(Y \leq uX~|~X>0) \cdot P(X>0) + P(Y \geq uX~|~ X<0) \cdot P(X<0) \\ &= \frac{ \int_0^{\infty}\int_0^{ux}f(x,y)dydx }{ P(X>0) } \cdot P(X>0) + \frac{ \int_{-\infty}^0\int_{ux}^0f(x,y)dydx }{ P(X<0) } \cdot P(X<0) \end{align} where like you said yourself, the integral expression $\int_{-\infty}^0\int_{ux}^0f(x,y)dydx = P(Y \leq uX)$ is unconditional and needs to be divided by $P(X>0)$.

Note that the starting point $P\left( \frac{Y}X \leq u \right)$ is unconditional, thus the final result should be unconditional as well. This couple of lines of derivation you see in the Wolfram page is just splitting the $X$-$Y$ plane into the disjoint regions via auxiliary conditioning.