Q$2)(30$p$)$ The permeability of a membrane used as a moisture barrier in a biological

application depends on the thickness of two integrated layers. The layers are normally

distributed with means of $0\mathrm{.}5$ and $1$ millimeters, respectively. The standard deviations of

layer thickness are $0\mathrm{.}1$ and $0\mathrm{.}2$ millimeters, respectively. The correlation between layers is

$0\mathrm{.}7$

$($a$)$ Determine the mean and variance of the total thickness of the two layers.

$($b$)$ What is the probability that the total thickness is less than $1$ millimeter?

$($c$)$ Let $x_1$ and $x_2$ denote the thickness of layers $1$ and $2,$ respectively. A measure of

performance of the membrane is a function $2x_1+3x_2$ of the thickness. Determine

the mean and variance of this performance measure.