How to compute probability of $(X = Y)$ for the joint pdf of $(x,y)$

Question

Let X and Y have joint probability density function $$f_{X,Y}(s,t) = 2e^{-(2s+t)},\qquad 0 \leq s, \ 0 \leq t$$ How to compute $Pr(X = Y)$, detailed explanation will be appreciated.

Answer 1

If there is a joint density for $(X,Y)$, that means that it is an absolutely continous vector. Then, for any (measurable) set $A\subset \mathbb R^2$, it is the case that $$P\big((X,Y)\in A\big)=\iint_A f(x,y) dxdy.$$

But since the set $A=\{(x,y)\in\mathbb R^2\colon x=y\}$ is a line, with no area, the double integral of any function over $A$ has to be zero. So $P(X=Y)=0$.