Suppose that \(X\) is a continuous random variable that takes on positive real values and has characteristic function \(\psi(t)=(1-\theta i t)^{-\alpha}\). Use Theorem 2.28 to find the density of \(X\).

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Theorem 2.28. Consider a random variable X with characteristic function (t). If L** \v(t)]dt < , then X has an absolutely continuous distribution F with a bounded and con- tinuous density f given by 1 f(x) = 2 exp(-itx)(t)dt.