# the composition of a random variable and its cdf

Let $$X$$ be a continuous random variable. Let $$F(t)=P(X\le t)$$ be the cdf (cumulative distribution function) of $$X$$. Then the random variable $$Y=F(X)$$ takes values in the unit interval $$[0,1]$$. What is the distribution of $$Y$$? I read from a book that seems to claim $$Y$$ has a uniform distribution. But I don't see why. $$Y$$ is too abstract for me to understand.

This is known as the probability integral transform. For $$0 we have \begin{align} F_Y(t) &= \mathbb P(Y\leqslant t)\\ &= \mathbb P(F_X(X)\leqslant t)\\ &= \mathbb P(X\leqslant F_X^{-1}(t))\\ &= F_X(F_X^{-1}(t))\\ &= t, \end{align} so that $$Y$$ is uniformly distributed over $$(0,1)$$. Note that when $$X$$ is not continuous the map $$F_X^{-1}$$ is not a true inverse, and must instead be defined as the quantile function $$F_X^{-1}(t) = \inf\{x:F_X(x)>t\}$$.