Here is a math problem:
Suppose we are told that and are given that . Show that .
Let’s start with the definition of expectation and use integration by parts.
since we are given .
Now using integration by parts, making the natural selection we have:
Recall that and plugging in our selections we get:
Let’s rewrite as . This simplifies to . Let’s plug this back into our equation above:
since plugging in to the first half of our expression just yields . We can’t actually evaluate at infinity, instead we take the limit:
, but recall that we are told and so we are simply left with:
, our desired result.