Let (X_{1}, X_{2}), and (X_{3}) be independent normal variables with [begin{array}{lll}Eleft(X_{1} ight)=5 & text { and }

Let \(X_{1}, X_{2}\), and \(X_{3}\) be independent normal variables with

\[\begin{array}{lll}E\left(X_{1}\right)=5 & \text { and } & \sigma_{1}^{2}=9 \\E\left(X_{2}\right)=-2 & \text { and } & \sigma_{2}^{2}=2.25 \\E\left(X_{3}\right)=5 & \text { and } & \sigma_{3}^{2}=4\end{array}\]

(a) Show that \(2 X_{1}+2 X_{2}+5 X_{3}\) has a normal distribution.

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