Calculating $\mathbb P(X=Y)$ for any continouous distribution or $X,Y \sim$ Negbinomial using symmetry

Question

Is it possible to calculate $\mathbb P(X=Y$) for continuous distribution such as $X,Y \sim\exp(\lambda)$, Negative binomial using symmetry

Using this calculation, we can calculate $\mathbb P(X<Y)$ and $\mathbb P(X\geq Y)$ by exchangeability?

Assume, $X,Y$ independent.

Answer 1

For the exponential case $P(X=Y)=0$ and $P (X<Y)=\frac 1 2$. This is true for any continuous distribution.

For negative Binomial distribution you have to calculate $P(X=Y)$ from the formula $P(X=Y)=\sum _n [P(X=n)]^{2}$. Once you do that you get $P(X<Y)=\frac 1 2 (1-P(X=Y))$. Of course, $P(X \geq Y)=P(X>Y)+P(X=Y)$.