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I have a test statistics $S(\theta_0) = $ number of $[X_i>0] $ that follows a binomial 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.

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