Let \(X\) and \(Y\) be independent normal random variables with

\[\begin{array}{lll}E(X)=4 & \text { and } & \sigma_{X}^{2}=25 \\E(Y)=3 & \text { and } & \sigma_{Y}^{2}=16\end{array}\]

(a) Use moment generating functions to show that \(5 X-4 Y+7\) has a normal distribution.

(b) Find the mean and variance of the random variable in part (a).