1. A production line produces two types of goods: A and B. The type of each...

Transcribed Image Text:

1. A production line produces two types of goods: A and B. The type of each produced good is chosen randomly (with probability 0.5 for each type), independently for each produced good. Each good of type A is defective with a probability of . Each good of type B is defective with a probability of . (a) Suppose that the production line is shut down immediately after 60 defective goods have been produced. i. Find the expectation for the total number of goods produced, proving any formula used (or giving a direct proof). ii. Find the standard deviation for the number of non-defective goods produced (you may cite a formula, if you wish). iii. State the Central Limit Theorem, justify why it can be used for part (iv), and explain why it might be useful for calcula- tions, in general. iv. Calculate the approximate probability that between 1100 and 1200 non-defective goods (inclusive) are produced. (b) Suppose that exactly 1200 goods are produced instead. Using Central Limit Theorem, with justification, find the probability that at least 70 defective goods are produced. Hint: Referring to a normal distribution table or calculator: (0.40) = 0.6554, (0.27) = 0.6064, (0.26) = 0.6026, (1.26) = 0.8962, (1.19) = 0.8832, D(1.32) = 0.9066, (1.39) = 0.9177 %3D %3D %3D 1. A production line produces two types of goods: A and B. The type of each produced good is chosen randomly (with probability 0.5 for each type), independently for each produced good. Each good of type A is defective with a probability of . Each good of type B is defective with a probability of . (a) Suppose that the production line is shut down immediately after 60 defective goods have been produced. i. Find the expectation for the total number of goods produced, proving any formula used (or giving a direct proof). ii. Find the standard deviation for the number of non-defective goods produced (you may cite a formula, if you wish). iii. State the Central Limit Theorem, justify why it can be used for part (iv), and explain why it might be useful for calcula- tions, in general. iv. Calculate the approximate probability that between 1100 and 1200 non-defective goods (inclusive) are produced. (b) Suppose that exactly 1200 goods are produced instead. Using Central Limit Theorem, with justification, find the probability that at least 70 defective goods are produced. Hint: Referring to a normal distribution table or calculator: (0.40) = 0.6554, (0.27) = 0.6064, (0.26) = 0.6026, (1.26) = 0.8962, (1.19) = 0.8832, D(1.32) = 0.9066, (1.39) = 0.9177 %3D %3D %3D