The probability that each item coming off a production line is defective is p and the...

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The probability that each item coming off a production line is defective is p and the probability that it is non-defective is q. 0<p<1, p + q= 1. At the beginning of a day's production, a quality control officer repeatedly inspects items each coming off a production line until he inspects n items. Let X be the number of defective items he finds. (a) Write down without proof the probability that X=k, indicating the possible values of k. Hence, by considering the expansion of [(p-q)]^n- [(q-p)] ^n. Show that the probability that X is even is 1/2[1+ [(1-2p)] ^n]. (Note that 0 is an even number) (b) Find the expected value of X, and write down without proof the variance of X. (c) If E(X) = 0.0125 and Var(X) = 0.9875, find to 4 decimal places the probability that X is odd. The probability that each item coming off a production line is defective is p and the probability that it is non-defective is q. 0<p<1, p + q= 1. At the beginning of a day's production, a quality control officer repeatedly inspects items each coming off a production line until he inspects n items. Let X be the number of defective items he finds. (a) Write down without proof the probability that X=k, indicating the possible values of k. Hence, by considering the expansion of [(p-q)]^n- [(q-p)] ^n. Show that the probability that X is even is 1/2[1+ [(1-2p)] ^n]. (Note that 0 is an even number) (b) Find the expected value of X, and write down without proof the variance of X. (c) If E(X) = 0.0125 and Var(X) = 0.9875, find to 4 decimal places the probability that X is odd.

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SOLUTION a The probability that X k denoted as PX k is given by the binomial probability formula PX k Cn k pk qnk where Cn k is the number of ways to choose k items out of n also known as n choose k o... View the full answer