A uniform inside a uniform

Idea. A uniform on $[0,X]$ is $X$ times an independent uniform on $[0,1]$. With $Y=XU$, every moment factors.

Step 1: moments. With $\mathbb{E}[X]=\mathbb{E}[U]=\frac{1}{2}$ and $\mathbb{E}[X^2]=\mathbb{E}[U^2]=\frac{1}{3}$:

• $\mathbb{E}[Y]=\frac{1}{2}\cdot \frac{1}{2}=\frac{1}{4}$;
• $\mathbb{E}[Y^2]=\frac{1}{3}\cdot \frac{1}{3}=\frac{1}{9}$, so $\mathrm{Var}(Y)=\frac{1}{9}-\frac{1}{16}=\frac{7}{144}$;
• $\mathbb{E}[XY]=\mathbb{E}[X^2]\mathbb{E}[U]=\frac{1}{6}$, so $\mathrm{Cov}(X,Y)=\frac{1}{6}-\frac{1}{8}=\frac{1}{24}$.

Step 2. With $\mathrm{Var}(X)=\frac{1}{12}$, $$\begin{array}{rl}\mathrm{corr}(X,Y)& =\frac{1/24}{\sqrt{\frac{1}{12}\cdot \frac{7}{144}}}=\frac{1/24}{\sqrt{7}/(24\sqrt{3})}\\ & =\sqrt{\frac{3}{7}}\approx \mathrm{0.655.}\end{array}$$

Remark. The relation is far from deterministic: even knowing $X$ exactly, $Y$ can be anywhere in $[0,X]$. Yet the correlation is fairly high, because a large $X$ allows large values of $Y$ and a small $X$ forces $Y$ to be small.