Question

A manufacturing company employs two devices to inspect output for quality control purposes. The first device is able to accurately detect 99.3% of the defective items it receives, whereas the second is able to do so in 99.7% of the cases. Assume that four defective items are produced and sent out for inspection. Let X and Y denote the number of items that will be identified as defective by inspecting devices 1 and 2, respectively. Assume that the devices are independent. Determine: a. fxy(x, y) b. fx(x) c. E(X) d. fy/2(y) e. E(Y | X = 2) f. V(Y | X = 2) g. Are X and Y independent?

Answer 1

Answer:

(a) fxy(x,y) = f(x), f(y)

(b) fx(x) = X = (99.3/100) x 4 = 3.972

(c) E(X) = 4

(d) fy/2y = (99.7/100) of (2 x 4) = 7.976

(e) E(Y | X = 2) = 2.

(f) V(Y | X = 2) = 2

Step-by-step explanation:

fxy(x,y) = f(x), f(y), as the question presented the assumption that the devices are independent, their measurements must also be independent

Since the first device is able to accurately detect 99.3% of the defective items it receives and the total defective items sent is 4,

(b) fx(x) = X = (99.3/100) x 4 = 3.972

(c) E(X) is expected result from first device. It is 4 because the items will be whole number.

(d) fy/2y = (99.7/100) of (2 x 4) = 7.976

(Since the first device is able to accurately detect 99.7% of the defective items it receives and the total defective items sent is 4)

(e) E(Y | X = 2) = 2. If the second device measure 2 defective items (X = 2), first device will also detect 2 because the measure 99.3% and 99.7% accurately. Their measurements will be approximately equal.

f. V(Y | X = 2). Like the explanation for (e) above, any relations V(Y | X = 2) will also be 2

(g) X and Y are independent because the are measure

measurements on two different devices