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How do I calculate expectations with the standard normal random variable? The integrals generally look very intimidating if even doable sometimes. I am curious how to approach these types of problems in general, so any advice is greatly appreciated. The problem in particular I am staring at right now is to find
, where is the standard normal random variable. The primary issue I'm having is how to calculate things like this--for instance, and referring to the exercise I mentioned, I write out the integral necessary to calculate this expectation, but it is the integral of the product of a double-exponential with a single-exponential.Offline
The function
Another way is to look at the moment generating function of a stardard normal random variable,
which is
Maths is fun!
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Thanks for your reply
I understand using the fact that
is odd to compute the mean, but my original question and what I still cannot figure out, is how to compute . As we have not approached moment generating functions in this course yet, I do not think I should use that route to find a solution. Any suggestions for my original problem?The function
is an odd integratable function
Therefore,
the expectation of a standard normal random variable
must equal 0.Another way is to look at the moment generating function of a stardard normal random variable,
which is
and
Last edited by almost there (2009-11-24 22:54:51)
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Ok, I feel foolish.
I meant what I said when I said "double exponential", buuuuuut I had wrongly interpreted the problem to be asking to evaluate the integral of the double exponential
when, in fact, the problem statement indicated I ought evaluate the integral , as you pointed out. Thank you, Avon.With a few solid nights' rest and a bit of hindsight, I can't imagine how I did not at some point realize that I wasn't even applying the definition of expectation correctly...what a goon.
If has the standard normal distribution then
which doesn't contain what I would call a double exponential (i.e. something like ).
Is this the integral that you are having problems with?
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