Consider two independent random variables: , where
( means uniform within support and .)
What is the following probability:
, where and are constants.
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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.