Let X 1 , . . . , X n be independent random variables each uniformly distributed
Question:
Let X1, . . . , Xn be independent random variables each uniformly distributed on [−1, 1]. Let pn = P(X21 + · · + X2n < 1). Conduct a simulation study to approximate pn for increasing values of n. For n = 2, p2 is the probability that a point uniformly distributed on the square [−1, 1] × [−1, 1] falls in the inscribed circle of radius 1 centered at the origin. For n = 3, p3 is the probability that a point uniformly distributed on the cube [−1, 1] × [−1, 1] × [−1, 1] falls in the inscribed sphere of radius 1 centered at the origin. For n > 3, you are in higher dimensions estimating the probability that a point in a “hypercube” falls within the inscribed “hypersphere.” What happens when n gets large?
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Solution When n gets large we can choose xi to be uniformly dis...View the full answer
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