We were doing order statistics in our probability class. And were asked to compute the following integral:
$$\iiint_{0 < x_{1} < x_{2} < x_{3} < 1} dx_{1}\, dx_{2}\, dx_{3}$$
I am unsure of how the limits of integration are being computed. Our professor told us that the answer would be $\int_{0}^{1} \int_{0}^{x_{3}} \int_{0}^{x_{2}}$. I just don't see why this has to be the case. Could someone please help me out a little?
$x_1$ ranges from $0$ to $x_2$. $x_2$ ranges from $x_1$ to $x_3$ but $x_1$ itself goes all the way down to $0$. So $x_2$ takes all values from $0$ to $x_3$. $x_3$ takes all values from $0$ to $1$. I hope this explanation helps.