Moment generating function of binomial distribution

by Math Avengers   Last Updated December 06, 2018 15:19 PM

I have a test statistics $S(\theta_0) = $ number of $[X_i>0] $ that follows a binomail distribution iwth $p=\frac{1}{2}$. With the standardized test statitics is $S=\frac{S(\theta_0)-(\frac{n}{2})}{\frac{\sqrt(n)}{2}}$, the solution shows that the moment generating function is $M_S(t) = [e^{-(t/2)/(\sqrt{n}/2)}*(\frac{1}{2}e^{t/(\sqrt{n}/2}+\frac{1}{2})]^n$.

My question is, shouldn't it simply be $[\frac{1}{2}*e^{t}+(1-p)]^n$ ?



Answers 1


You do not get the Binomial MGF, since your standardised statistic is not binomial but something else. When you subtract

$S(\theta_0) - \frac{n}{2}$

you are shifting the binomial random variable $S(\theta_0)$ values to the left.

However, binomial random variable is always positive, while your shifted statistic can become negative, so it is not binomial anymore.

SWIM S.
SWIM S.
December 06, 2018 15:10 PM

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