TITLE: Multiple random varialbles
QUESTION [1 upvotes]: $$f_X(x_1,x_2,x_3) = \begin{cases}
         \frac{1}{x_1x_2} &  0\leq x_3  \leq x_2 \leq x_1 \leq 1;\\[8pt]
        0 & \text{otherwise} \end{cases}  $$
question: Are $x_1,x_2,x_3$ are independent random variables.
to prove independence I have find individual distribution  of random variable in term of other two random variables.
But how to calculate it .How to put limits if I am calculation $f_{x_1}(x_1)$

REPLY [0 votes]: The mere fact that $x_2$ is constrained to be less than $x_1$ tells you that they're not independent.
(And you shouldn't be calling the random variables by the same names as the arguments to the density function.  If you write $f_{X_1,X_2}(x_1,x_2)$, using capital letters for the random variables and lower-case letters for the arguments, then you can make sense of such expressions as $f_{X_1,X_2}(0.8,0.3)$ or $\Pr(X_1 < x_1)$.)
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