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August 19, 2021

Computational Example on Probability Distribution

math

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