Prove Theorem 2.32. That is, suppose that \(X\) is a random variable with characteristic function \(\psi(t)\). Let \(Y=\alpha X+\beta\) where \(\alpha\) and \(\beta\) are real constants. Prove that the characteristic function of \(Y\) is \(\psi_{Y}(t)=\exp (\) it \(\beta) \psi(\alpha t)\).

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Theorem 2.32. Suppose that X is a random variable with characteristic function (t). Let Y = aX+ where a and are real constants. Then the characteristic function of Y is by (t) = exp(it)(ta).