# Distribution function, applied to itself? [duplicate]

Consider random variable $X$ with continuous, increasing CDF $F_X (x)$. Let $W=F_X (X)$. Characterize the distribution of $W$.

I get $F_W (w)=\mathrm{Pr}(F_X (X) \leq w)=\mathrm{Pr}(X\leq F_X^{-1}(w))=F_X(F_X^{-1}(w))=w$

Can I just impose that $w=1$ and say this is a degenerate distribution? That seems off to me because the question doesn't actually specify a domain for $x$. Did I miss something?

## marked as duplicate by whuber♦Jan 28 '14 at 4:19

• Note that $W=F_X(X)\in [0,1]$, and you proved that $P(W\leq t)=F_W(t)=t$. Hence, $W\sim \mathrm{U}[0,1]$. – Zen Jan 28 '14 at 2:27
$W$ follows the uniform distribution on $[0,1]$ because the image of $F_{X}(X)$ is the closed interval $[0,1]$ (which is in turn the domain of $F_{W}$).