|February 12th, 2012, 05:38 PM||#1|
Joined: Apr 2011
Error with Moment Generating Functions
Just a warning, I may use improper terms here. If you think I'm explaining something wrong, it's quite possible that I am. Just let me know.
I'm trying to calculate the second moment for a binomial distribution. E(y^2), where Y ~ Binom.
I have two different methods, both of which I believe should work, but I'm getting different results. Could someone help me find which one I've done incorrectly?
V(Y) = E(y^2) - E(y)^2
==> npq = E(y^2) - (np)^2
==> E(y^2) = np(q + np)
The second method I'm using is with the second moment of the moment generating function.
My thinking is that if I take the MGF for the binomial distribution, take its derivative, and set t = 0, I should get E(y^2)
mgf = (p*e^t + q)^n
mgf(first derivative) = n*p*e^t*(p*e^t + q)^(n-1)
mgf(first derivative, t = 0) = n*p*(p + q)^(n-1)
mgf(first derivative, t = 0) = n*p
Obviously these both can't be right. I'm thinking the first one is correct, but where did I mess up?
|February 13th, 2012, 03:18 PM||#2|
Joined: May 2007
Re: Error with Moment Generating Functions
Your second method (moment generator) is correct. However what you got is the first moment (mean), not the second moment. You need to use the second derivative to get the second moment.
|error, functions, generating, moment|
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