Question

A certain raw material is classified as to moisture content X (in percent) and impurity Y (in percent). Let X and Y have the joint pmf given by:

X
Y	1	2	3	4
2	.10	.20	.30	.05
1	.05	.05	.15	.10

a) Find the marginal pmf's, the means and the variances.

b) Find the covariance and the correlation coefficient of X and Y.

c) If additional heating is needed with high moisture conent and additional filtering with high impurity such that the additional cost is given by the function C =2X + 10Y^2 in dollars, find E(C).

I have the answers from the textbook below, but I need to see the steps in getting to these answers. Thanks!

~~~~~~~~~~~~~~~~~~~~~~~~~~~

Answers:

a) f1 (1) = 0.15, f1(2) = .25, f1(3) =.45 , f1(4) = .15

f2(1) = .35, f1(2) = 0.65

,, ,

b) Cov (X,Y) = -.0900 ,

c) E (C) = 34.70

Answer 1

PMF of x 2 3 4 Total PCX) XP(X) XA2PX)0.15 0.15 0.25 0.45 0.15 0.15 0.5 1.35 1 4.05 0.6 2.6 2.4 7.6 Mean(X)-XP(X)-2.6 Varzince= 7.6-2.62-0.84 PMF of Y 1 Total 2 0.65 0.35 1 P(Y) Y*P(Y) YA2P 1.3 0.35 1.65 2.6 0.35 2.95 Mean(Y)- yYP(Y) 1.65 Varaince -2.05-1.350.2275 0.4 0.4 2*1*0.1-0.2 1.8 0.05 1*2*0.05-0.1 0.45 0.8 4.2 Co var iance -E(XT)-E(X)EY) 4.2-2.6(1.65)-0.09 -0.09-= 0.2059 Correlation - f0.84002215) E(C) 2E(X)+10E) 2 2.6-10 2.9524.3