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(Solved): 5-93. The geometric random variable \( X \) has probability distribution \[ f(x)=(1-p)^{x-1} p, \q ...



5-93. The geometric random variable \( X \) has probability distribution
\[
f(x)=(1-p)^{x-1} p, \quad x=1,2, \ldots
\]
Show t

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 \( X \).


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