Uniform Distribution Problem

Question

Let $X$ be a random variable uniformly distributed in $[0,1]$, and let $Y$ be a RV uniformly distributed in $[X,1]$. I want to calculate the theoretical distribution of $Y$, any hints? I already tried with simulation, which gave me an idea of what $Y$ looks like, but nothing more.

Answer 1

I would try a conditioning argument: for $0<y<1$ \begin{align} P[Y \leq y] &= {\bf E}[ P[Y \leq y \mid X]] \\ &= {\bf E}\left[ \frac{y-X}{1-X} I_{[X \leq y]} \right] \\ &= \int_0^y \frac{y-x}{1-x}dx \\ &= \left. -(y-1)\ln(1-x) + x \right|^y_0 \\ &= -(y-1)\ln(1-y) + y. \end{align}

Answer 2

$$P(Y<y)=\int_x P(Y<y|X=x) P(X=x) dx=\int_0^y \frac{y-x}{1-x} dx=[(1-y)\ln(1-x)+x]_{x=0}^{x=y}=(1-y)\ln(1-y)+y,$$ so the density function is its derivative $f_Y(y)=-\ln(1-y)$ for $y\in(0,1)$.