As a review of statistics I try to work on this problem:
Problem. Let be a 3-dimensional random variable which follows the distribution (i.e., the probability density function is)
Find the probability density function of .
I start off by guessing the answer to be very sloppily, which is of course wrong and I tried to figure out how I can relate with .
Solution
Let us start from the definition, what does tell us? In view of a distribution it tells us how are 's spreaded in . In other words, for every given , we have
which is the proportion of 's lying within and denotes the Lebesgue measure on . From that recall also that to find the probability density function of , it is sufficient to find its cummulative distribution (since then we can differentiate pointwise).
From this, consider the relation , which is
we conclude that () if and only if lies in the set
therefore
The answer is