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 0.5 and 1 millimeter respectively. The standard deviations of layer thickness are 0.1 and 0.2 millimeters, respectively. The correlation between layers is 0.7.
(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 X1 and X2 denote the thickness of layers 1 and 2, respectively. A measure of performance of the membrane is a function 2X1 + 3X2 of the thickness. Determine the mean and variance of this performance measure.