Suppose that \(X\) is a discrete random variable that takes on positive integer values and has characteristic function

\[\psi(t)=\frac{p \exp (i t)}{1-(1-p) \exp (i t)}\]

Use Theorem 2.29 to find the probability that \(X\) equals \(k\) where \(k \in\) \(\{1,2, \ldots\}\).

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Theorem 2.29. Consider a random variable X that takes on values in the set {kd+1kZ} for some d > 0 and 1 R. Suppose that X has characteristic function (t), then for any r = {kd+1: k = Z}, d /d P(X = x)= = exp(-itx)(t)dt. 2T