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5-93. The geometric random variable \( X \) has probability distribution \[ f(x)=(1-p)^{x-1} p, \quad x=1,2, \ldots \] Show that the moment-generating function is \[ M_{X}(t)=\frac{p e^{t}}{1-(1-p) e^{t}} \] Use \( M_{X}(t) \) to find the mean and variance of \( X \).

Ans: the geometric random variable X has the probability distri