Prove Theorem 2.24. That is, suppose that \(X\) is a random variable with moment generating function \(m_{X}(t)\) that exists and is finite for \(|t|0\). Suppose that \(Y\) is a new random variable defined by \(Y=\alpha X+\beta\) where \(\alpha\) and \(\beta\) are real constants. Prove that the moment generating function of \(Y\) is \(m_{Y}(t)=\exp (t \beta) m_{X}(\alpha t)\) provided \(|\alpha t|

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Theorem 2.24. Let X be a random variable with moment generating func- tion mx(t) that exists and is finite for |t| 0. Suppose that Y is a new random variable defined by Y = aX+ where a and are real con- stants. Then the moment generating function of Y is my (t) = exp(t)mx(at) provided |at| < b.