How to find $P(X<Y)$ where $X$ and $Y$ are random variables?

Question

I'm trying to understand this proof in this book page 319:

My question is why $f_{T_1,T_2}(t_1,t_2)$ is the circled area above? in fact, my question is more general. I'm having troubles to understand $P(X<Y)$ where $X$ and $Y$ are random variables. What I know if $P(X<x)$. I'm very confused, please help.

Answer 1

The joint PDF is the PDF of the random vector $(T_1,T_2)$.

"Morally" it assigns to each point $(x,y)$ the "mass" of that point, i.e. the "probability" that $(T_1,T_2)$ is equal to $(x,y)$.

The precise formulation of this fact is that the PDF $f_{(T_1,T_2)}$ satisfies $$\mathsf P((T_1,T_2)\in A)=\int_{A} f_{(T_1,T_2)}(x,y)\,\mathrm d(x,y)$$ for all measurable $A\subset\mathbb R^2$. We are interested in the set $A=\{(t_1,t_2)\in\mathbb R^2 : t_1<t_2\}$, because with that choice, $T_1<T_2$ if and only if $(T_1,T_2)\in A$. So the book computes

$$\int_{A} f_{(T_1,T_2)}(t_1,t_2)\,\mathrm dt_1\,\mathrm d t_2.$$

Answer 2

Whenever two random variable are independent , their joint density is simply the product of their marginal densities . Here X and Y are independent . So

f(x,y) =f(x) * f(y)

Now . P(X<Y) = P(X<y| Y=y)

Now for X we need to integrate from 0 to y ....but Y Can take any value from 0 to infinity .....So need to integrate over the whole range of Y .