Let $X_1 \sim Uniform(0,1)$, and $X_2 \sim Uniform(0, x_1)$, where $x_1$ is the realized value of $X_1$. Find $P(X_1 + X_2 \geq 1)$.
I know that I need the joint distribution of $X_1$ and $X_2$.
$f_1(x_1) = 1, \ 0 < x_1 < 1$
$f_{2|1}(x_2|x_1) = \frac{1}{x_1}, \ 0 < x_2 < x_1$
$f_{1,2}(x_1, x_2) = f_{2|1}(x_2|x_1)f_1(x_1) = \frac{1}{x_1}, \ 0 < x_2 < x_1 < 1$.
What are the limits of integration when integrating this joint PDF to get the desired probability?