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 $$[\frac{1}{2}*e^{t}+(1-p)]^n$$ ?

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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.
December 06, 2018 15:10 PM