Consider two independent random variables: , where
and .
( means uniform within support and .)
What is the following probability:
, where and are constants.
Any suggestions, tips or ideas?
Hello,
Since they're independent, their joint probability density function is
Now, (1 is the indicator function)
By the law of the unconscious statistician, which states that for any measurable function f,
where g is the joint probability function of
So here, we have :
now, has to be interpreted as a region.
Since we first integrate with respect to r, consider it with respect to r :
So finallly :
Note : the inner integral doesn't necessarily goes to infinity, it depends on the support of the rv r, which will be expressed in the function f.