Let's say we have two random variables $T_1$ and $T_2$ and the joint density function of the two is uniform over the region $0\leq t_1\leq t_2 \leq L$, where L is a positive constant. Then the area of the region would equal $\frac{L^2}{2}$.
Now, I need to find the expected value of the sum of the squares, $E[T_1^2+T_2^2]$, which could easily be found by finding the density function, $f(t_1,t_2)$, and computing a double integral of $(t_1^2+t_2^2)\cdot f(t_1,t_2)$ while setting the bounds over the region of the triangle $0\leq t_1\leq t_2 \leq L$.
I'm told the density function here is $\frac{2}{L^2}$. Is the density function of 2 uniformly distributed random variables $\frac{1}{Area}$, just as the density function of 1 uniformly distributed random variable is $\frac{1}{Distance}$?
If so, when dealing with 3 variables will $f(T_1,T_2,T_3) = \frac{1}{Volume}$?
Is there a general rule for an $n$ amount of variables?