Given the moment-generating function $Mx(t) = e{^{3t+8t{^2}}},$ find the moment generating function of the random variable $Z = {\frac{1}{4}}(X - 3)$, and use it to determine the mean and variance of Z.
2. $M_{aX+b} = e^{bt}M_{X}(at)$
Then $M'_{aX+b}(0) = \mu$ and $E[Z^2] = M''_{aX+b}(0)$ and so $\text{Var}(Z) = E(Z^2) - \mu^2$.