Question

let x denote the number of major defects for a particular piece of machinery and y be the number of cosmetic flaws on this same piece, suppose that x and y are independent variables with f_1(x) = .80, .15, and .05 for x = 0, 1, 2 respectively and f_2(y) = .50, .25, .15, .08, .02 for y = 0, 1, ..., 4 respectively. a) what is the joint mass function of these two variables? b) what proportion of these machines will have no major defects or cosmetic flaws? what proportion will have at least one defect or flaw? c) for what proportion of these machines will the number of cosmetic flaws exceed the number of major defects?

Answer 1

a) The joint mass function of x and y is f(x,y) = f1(x) * f2(y). b) The proportion of machines with no major defects or cosmetic flaws is 0.40, and the proportion with at least one defect or flaw is 0.60. c) The proportion of machines where the number of cosmetic flaws exceeds the number of major defects is 0.55.

Step-by-step explanation:

a) The joint mass function of two variables is the product of their individual mass functions when the variables are independent. In this case, the joint mass function of x and y would be f(x,y) = f1(x) * f2(y). So, for example, when x = 0 and y = 0, the joint mass function would be f(0,0) = f1(0) * f2(0) = 0.80 * 0.50 = 0.40.

b) To find the proportion of machines with no major defects or cosmetic flaws, we need to find the joint probability when x = 0 and y = 0. So, f(0,0) = 0.40. Similarly, to find the proportion of machines with at least one defect or flaw, we need to find the sum of joint probabilities when x > 0 or y > 0. So, P(X > 0 or Y > 0) = 1 - P(X = 0 and Y = 0) = 1 - 0.40 = 0.60.

c) To find the proportion of machines where the number of cosmetic flaws exceeds the number of major defects, we need to sum the joint probabilities for y > x. So, P(Y > X) = f(0,1) + f(0,2) + f(1,2) + f(0,3) + f(1,3) + f(2,3) + f(0,4) + f(1,4) + f(2,4) + f(3,4) = 0.15 + 0.05 + 0.25 + 0.08 + 0.02 = 0.55.