# Given that x~U[0,1] and y~U[0,1], derive the conditional CDF of W=x-b*(x-y)^2 where 0<b<1? Condition on x (i.e. treat x as a constant).

Given that $$x\sim U[0,1]$$ and $$y\sim U[0,1]$$, derive the conditional CDF of $$W=x-b\cdot(x-y)^2$$ where $$0? Condition on $$x$$ (i.e. treat $$x$$ as a constant). I am running into difficulties with this, given the two to one transformation in parts.

• Conditional on what exactly and what is the meaning of the trailing phrase “where 0”? – Nap D. Lover May 1 at 1:48

\begin{align}\mathsf P(W\leq w\mid x=s)&=\mathsf P(x-b(x-y)^2\leq w\mid x=s)\\[2ex]&=\mathsf P\left(s-b(s-y)^2\leq w\right)&\text{independence of }x,y\\[2ex]&=\mathsf P\left(\tfrac{s-w}b\leq (s-y)^2\right)&\text{since }0
NB: Recall that: $$\{c\leq Z^2\}=\{Z\leq -\surd c\}\cup\{ \surd c\leq Z\}$$