Calculating $\text{Pr}(X_1<X_2)$

Question

I have two independent random variables $X_1$ and $X_2$ with probability density functions $f_1(x)$ and $f_2(x)$, respectively, how can I calculate the probability $\text{Pr}(X_1<X_2)$?

Answer 1

If the random variables are independent, then their joint distribution is $f_{X_1,X_2}(x_1,x_2)=f_1(x_1)f_2(x_2)$. Then the required probability is $$P(X_1<X_2)\\=\int_{-\infty}^{\infty}\int_{-\infty}^{x_2}f_{X_1,X_2}(x_1,x_2)dx_1dx_2=\int_{-\infty}^{\infty}f_2(x_2)\left[\int_{-\infty}^{x_2}f_1(x_1)dx_1\right]dx_2$$

Answer 2

If you knew $X_2=x_2$, the answer would be $\int_{-\infty}^{x_2}f_1(x_1)dx_1$. Since you don't, independence implies the answer is $\int_\mathbb{R}f_2(x_2)\int_{-\infty}^{x_2}f_1(x_1)~dx_1~dx_2$.