Uniform distribution dependent on another uniform distribution

Question

I was asked this question on an interview and had trouble so I'm turning here after the fact for help understanding what I should have done. Suppose we choose $x$ uniformly on $[0,1]$. Now we choose $y$ uniformly on $[x,1]$. What's the distribution of $y$? Clearly $y$ could be anything on $[0,1]$ since we could have $x=y$. But I think the higher values of $y$ should be more likely since there are more choices of $x$ for which they are possible, so I don't think $y$ is uniformly distributed. But how do I calculate what it's distribution actually is?

Answer 1

We just have use total law of probability:\begin{align} P_Y(y) &= \int_0^1 p_{Y|X(y|x)}p_X(x) \,dx \\ &=\int_0^y \frac{1}{1-x}\, dx \end{align}