How can we work with a joint probablility density function with more than 3 variables?

Question

This is basically more of a theoretical question and nothing I actually encountered. I have been thinking that is totally possible to need more than $3$ variables to represent a joint probability density function, in class I have learned only the two variables version where we simply do a double integral, and it is quite obvious to see that for 3 variables we can do a triple integral but what happens when we have more than three? We can't possibly draw or imagine anything in the fourth dimension fo set up the integral limits in thr first place. I know we don't need to always draw to get the correct limits but sometimes it is the case. Any help would be appreciated.

Answer 1

In principle, "drawing" the integration boundaries is just an assist to the mind, and interchanging integration boundaries can be done formally, without need of sketches. When you reach the point in your studies that you need to integrate over multiple variables with tricky dependencies between the integration limits, you will probably have enough technical proficiency to deal with it.

If you are still looking for a method to sketch high dimension integration boundaries, you can try picking pairs of integration variables and sketch the relationship between each pair. That is, given a complicated $\int_{f_1(x_2 \cdots x_n)} dx_1\cdots \int_{f_n(x_1\cdots x_{n-1})} dx_n$ you can try drawing the boundaries of pairs $x_a;x_b$, keeping all other variables fixed.