The question I have is:

Define X,Y to be two independent uniform(0,1) random variables and $Z:=\frac{Y}{X}$

Compute  $P(X<x|\sigma(Z))$

The answer given apparently by "straightforward elementary computations" is for $x\geq 0$

$P(X<x|\sigma(Z))$=min$\{{x^2,1}\}\mathbb{I}_{Z\leq 1}$ $+$ min$\{{x^2 Z^2,1}\}\mathbb{I}_{Z\geq 1}$.

My idea was to condition on the $Z\leq 1$ and ${Z\geq 1}$ then compute using the joint density of X and Y, but this seems to work for the first term but doesn't for the second? Any help would be much appreciated.

The question I have is:

Define $X,Y$ to be two independent uniform(0,1) random variables and $Z:=\frac{Y}{X}$

Compute  $P(X<x|\sigma(Z))$.

The answer given apparently by "straightforward elementary computations" is for $x\geq 0$,

$$P(X<x|\sigma(Z)) = \min\{{x^2,1}\}\mathbb{I}_{Z\leq 1} + \min\{{x^2 Z^2,1}\}\mathbb{I}_{Z\geq 1}.$$

My idea was to condition on the $Z\leq 1$ and ${Z\geq 1}$ then compute using the joint density of $X$ and $Y$, but this seems to work for the first term but doesn't for the second? Any help would be much appreciated.