Problem 6. An industrial process manufactures items that can he classified as ei
ID: 3277745 • Letter: P
Question
Problem 6. An industrial process manufactures items that can he classified as either defective or not defective. The probability that an item is defective is 0.1. An experiment is conducted in which 3 items are drawn randomly from the process. Let X indicate whether the last item drawn is defective (i.e., X 1 if the last item is defective and X -0 otherwise), and let Y denote the number of defective items in the sample. (a) Find the joint p.m.f. of X and Y (b) Find the marginal distributions of X and Y (c) Find the probability that at least one item in the sample is defective.Explanation / Answer
a) Here the joint PDF of X, Y is computed as:
b) Now the required marginal distribution of X is computed by simply adding the columns in the above table to get:
P(X =0 ) = 0.729 + 0.162 + 0.009 = 0.9 and P(X =1 ) = 0.081 + 0.018 + 0.001 = 0.1
Similarly, the marginal PDF for Y is computed as:
P(Y=0) = 0.729, P(Y=1) = 0.162 + 0.081 = 0.243, P(Y=2)= 0.009 + 0.018 = 0.027 and P(Y=3) = 0.001
c) The probability that at least one item in the sample is defective is computed as:
= 1- Probability that no item is defective.
= 1 - P(Y=0)
= 1 - 0.729
= 0.271
Therefore 0.271 is the required probability here.
X=0 X=1 Y=0 0.9*0.9*0.9 = 0.729 0 Y=1 2* 0.1*0.9*0.9 = 0.162 0.081 Y=2 0.1*0.1*0.9 = 0.009 2*0.009 = 0.018 Y=3 0 0.1*0.1*0.1 = 0.001