Example (Practice Exercise 5.8)

Find $\mathbb{P}[Y > y]$ where : $$X \sim Uniform[1,2]\\ Y\mid X \sim Exponential(x)$$

We have then $f_{Y\mid X} = x e^{-x y}$ (the Exponential distribution with rate x). For a given X, then we can compute $\mathbb{P}[Y > y \mid X= x]$:

$$\mathbb{P}[Y> y \mid X= x] = \int_{y}^{\infty}x e^{-xy'}dy' = e^{-xy}$$ and we can now integrate over the entire ($\Omega_X$) distribution for x to get the final answer:

$$\begin{align} \mathbb{P}[Y> y] &= \int_{\Omega_X}\mathbb{P}[Y> y \mid X= x] f_X(x) dx \\ &= \int_{1}^{2} e^{-xy} dx \\ &= \frac{1-e^{-y}}{y} \end{align}$$