Discussion
The distribution with moment generating function \(\dfrac{1}{3 - 2e^{t}}\) is :
Answer: C
The distribution is the geometric distribution.
Rewrite the m.g.f.:
\[ \begin{aligned} \frac{1}{3 - 2e^{t}} &= \frac{1/3}{1 - \frac{2}{3}e^{t}} \\ &= \sum_{k=0}^{\infty} \frac{1}{3}\left(\frac{2}{3}\right)^{k} e^{tk} \end{aligned} \]
for \(e^{t} \lt 3/2\). This is the m.g.f. of \(P(X = k) = \frac{1}{3}(\frac{2}{3})^{k}\), \(k = 0, 1, 2, \ldots\), a geometric distribution with \(p = 1/3\), whose m.g.f. is \(p/(1 - qe^{t})\).
The exponential has m.g.f. \(\lambda/(\lambda - t)\), with \(t\) not \(e^{t}\); the Cauchy distribution has no m.g.f.; and the logarithmic series m.g.f. involves \(\log(1 - qe^{t})\).
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