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N + r)(m - r)' Also, determine the probabilities P(z ~ Po > p) and P(z ~ P1 < p) for fixed Po and Pl' 42 EXERCISES IN PROBABILITY AND STATISTICS 67 If x and yare the maximum values obtained from independent samples of 111 and 11 (m ~ 11) observations respectively from a rectangular distribution in the (0, 1) interval, find the sampling distribution of the product z = Xy in the two cases (i) m = nand (ii) m < n. , but that for m < n the rth moment about the origin of v is E(v r ) = G),. ), ().

Hence derive the expectation of the total score (Sn) of the forecaster for the n days, and show that this attains its maximum value for A. = 1. ], and thereby deduce that, for fixed n, this variance is a maximum for A. = t. 98 In testing the quality of large batches of a mass-produced article, randomly selected items of the product are examined individually from each batch, the probability of finding a defective item in a batch being a constant p, (0 < p < 1), and that of a non-defective q (= 1- pl.

Y) - (ex+3)(ex+2)2' (P+ 1)x 2 = (P+ l)x. X = x) = (P+3)(P+2)2' P+2 • and determine the correlation between X and Y. Comment on the limiting forms of these two conditional distributions for Y = 1 and X = 0 respectively. 66 If x is the largest of m independent observations from a uniform distribution in the interval (0, ex), and y the largest observation in a random sample of n from a uniform distribution defined in the range (0. ,----, ·z11-1 dz -:(m+n)p" for 0 ~ z ~ p, ' and mnpm dz -'m+n zm+1' ~or I' p ~ ~ z< 00.

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