Question

Let \(X\) be a normal random variable with mean \(\mu\), variance \(\sigma^{2}\), pdf

$$ f(x)=\frac{1}{\sigma \sqrt{2 \pi}} e^{-\frac{(x-\mu)^{2}}{2 \sigma^{2}}} $$

and mgf \(M(t)=e^{\mu t+\frac{1}{2} \sigma^{2} t^{2}}\)

(a) Prove, by identifying the moment generating function of \(a+b X\), that \(a+b X \sim\) \(N\left(a+b \mu, b^{2} \sigma^{2}\right)\)

(b) Prove, by identifying the pdf of \(a+b X\) (via the cdf), that \(a+b X \sim N(a+\) \(\left.b \mu, b^{2} \sigma^{2}\right)\)