Random variable is parameter for distribution of another random variable

Question

What would you do to find probability when a uniform random variable is the parameter for the distribution of another uniform random variable.

ie: $Z \sim Unif(0,1)$ $Y \sim Unif(0,Z)$

And we are interested in finding the probability of $Y$ either $>, <$, or $=$.

I feel as if I have to use a double integral but am not sure where to go.

Answer 1

Guide:

Use conditional probability, first figure out $P(Y \le y|Z=z)$ and $f_Z(z)$

Now you can compute the following:

\begin{align} P(Y \le y) &= \int_y^1 P(Y \le y|Z=z)f_Z(z)\, dz \end{align}