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#1 2009-11-24 05:22:46

almost there
Member
Registered: 2009-11-11
Posts: 21

expectation of standard normal random variable

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.

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#2 2009-11-24 11:10:29

gckc123
Member
Registered: 2009-10-19
Posts: 15

Re: expectation of standard normal random variable

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


Maths is fun!

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#3 2009-11-24 22:11:51

almost there
Member
Registered: 2009-11-11
Posts: 21

Re: expectation of standard normal random variable

Thanks for your reply smile

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?



gckc123 wrote:

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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#4 2009-11-25 09:05:52

Avon
Member
Registered: 2007-06-28
Posts: 80

Re: expectation of standard normal random variable


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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#5 2009-12-03 07:11:24

almost there
Member
Registered: 2009-11-11
Posts: 21

Re: expectation of standard normal random variable

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.


Avon wrote:


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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