What is $P(X<Y)$ when $X,Y$ represent a random sample from two distributions.

Question

The income of people in two cities is represented by two Pareto-type pdfs: $$f(x)=\frac{2}{x^3},1<x<\infty,g(y)=\frac{3}{y^4},1<y<\infty$$ One person is selected at random from each city. Let $X$ and $Y$ be their respective incomes. Compute $P(X<Y)$.

It makes sense to me that this would be the integral of $f(x)$ when it is less than $f(y)$ but I am not sure how to set this up. Perhaps there is a better way altogether? Can I have a hint?

Answer 1

Assuming $X$, $Y$ are independent, the desired probability is given by $$\Pr[X < Y] = \int_{x=1}^\infty \int_{y=1}^\infty \mathbb{1}(x < y) f_{X,Y}(x,y) \, dy \, dx = \int_{x=1}^\infty \int_{y=x}^\infty f_X(x)f_Y(y) \ dy \, dx $$ where $$\mathbb{1}(x < y) = \begin{cases} 1, & x < y \\ 0, & \text{otherwise}. \end{cases}$$

Answer 2

By the partition theorem;

$$\mathbb{P}(X < Y) = \int_Y \mathbb{P}(X < Y \mid Y = y)g(y)~dy$$