I have two i.i.d continues random variables $X_0, X_1$, with PDF $f$, where the support $subset$ $(0,infty)$.

I need to calculate $P(X_0 geq X_1)$. I know the answer is $frac{1}{2}$ due to symmetry. However, I want to directly prove it using integrals.

My try:

We know that $f(x,y) = f(x)f(y)$.

So need to show:

$int_{0}^{infty} int_{0}^{x_0}f(x_1)f(x_0),dx_1,dx_0 = frac{1}{2}$

Since $f(x,y) = f(x)f(y)$ is a PDF we have:

$1 = int_{0}^{infty} int_{0}^{infty}f(x_1)f(x_0) ,dx_1,dx_0 = int_{0}^{infty} int_{0}^{x_0}f(x_1)f(x_0) ,dx_1,dx_0 + int_{0}^{infty} int_{x_0}^{infty}f(x_1)f(x_0) ,dx_1,dx_0 $

So it is enough to show that:

$int_{0}^{x_0}f(x_1)f(x_0) ,dx_1,dx_0 = int_{0}^{infty} int_{x_0}^{infty}f(x_1)f(x_0) ,dx_1,dx_0 $

To conclude. I tried manipulating the integral, but I wasn’t able to show it. I think I need to use Fubini, but I am not sure.

Thanks for the help!