Here's a probability question I feel like I should know the answer to. Let $X\sim U(0,1)$ and for each sample $x$ let $y$ be chosen from $U(0,x)$. How is $Y$ distributed?

My naive guess had been a linear distribution but that doesn't seem to match what I get if I simulate in Mathematica

Figured it out: $Y$ has the pdf $p_Y(y)=\log(1/y)$.

(with support on $y\in (0,1]$)

Nice@Semiclassical!

@SangchulLee I was wondering, do you have any recommendations (problem books, textbooks) for the subjects real analysis, measure theory and probability ?

@GabrielRomon Sheldon Ross' A first course in Probability is my favourite as an introductory text to Probability.