Distribution of a conditional random variable defined from two independent random variables

Question

Let $X$ and $Y$ be independent real-valued random variables with distributions $f_X$ and $f_Y$ respectively.

Consider a new random variable $Z=(X\mid X>Y)$.

It is defined as $Z=X$ on a subset $\Omega':=\{\omega\mid X(\omega)>Y(\omega)\}$ of the sample space $(\Omega,P)$, and the probability measure on $\Omega'$ is normalized as $P'(A)=P(A)/P(\Omega')$ for measurable $A\subset\Omega'$.

How can I describe the distribution of $Z$?

Answer 1

We can directly describe $Z$ using a pdf $f_Z$. To do this, notice that,

$$P(\Omega’) = \int_{\mathbb{R}}\int_{(-\infty,x)}f_X(x)f_Y(y)\,dy \,dx$$

If $P(\Omega’)=0$, then $Z$ is not well-defined. Otherwise,

$$P(Z\in A)= \frac{\int_A \int_{(-\infty,x)} f_X(x)f_Y(y)\,dy\,dx}{P(\Omega’)}.$$

Differentiating and applying the fundamental theorem of calculus yields,

$$f_Z(z) = f_X(z)\frac{\int_{(-\infty,z)} f_Y(y)\,dy}{\int_{\mathbb{R}}\int_{(-\infty,x)} f_X(x)f_Y(y)\, dy\, dx}$$