8.The proportion X1 and X2 of two chemicals found in samples of an insecticide have the joint probability density function f(x, y) = 2, 0 X 1, 0 Y 1, 0 X + Y 1

The random variable W = X1 + X2 denotes the proportion of the insecticides due to the combination of both chemicals. Prove that the variance of W is one of the choices below. Choose one and prove.

(a) 1/2

(b) 1/18

(c) 2/3

(d) 0

Prove how you get answer in the space below. Start with Var(W) = E(X + Y E(X + Y))^2 = double integral (x + y E(x + y))^2*f(x, y)dx dy and complete the integral. Show all intermediate work.

9. X, The temperature (0C) of the bottom O-rings in a space shuttle in orbit and Y, the temperature of the upper O-rings in the same space shuttle, are believed to follow a bivariate normal distribution with parameters x = 200, x = 15, = 0.7, y = 250, y = 30

What is the variance of the total temperature of the O-rings (upper and lower combined)? Choose one and show work.

(a) 1125

(b) 1755

(c) 33.54

(d) 41.89