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abigail018
abigail018
30.07.2021 • 
Mathematics

Consider an urn initially containing n balls, numbered 1 through n, and suppose that balls will be randomly drawn from the urn, one by one, and without replacement (so that after n draws, it is empty'). Letting X be the number of successes that will occur, where a success is considered to occur on the ith draw if the ball obtained is numbered i or smaller, give the expected value of X, expressed as a fairly simple function of n. (E.g., if n = 5, and the balls are drawn in the order 3, 1, 5, 4, 2, then r = 3, because the 2nd, 4th, and 5th draws result in successes, but the 1st and 3rd draws don't.) (Hint: Express X as a sum of simple random variables which can only assume the values 0 and 1.)

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