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U = (Y − µ Y) − b(X−µx), where b is a fixed constant that I choose It follows that U2 = (Y − µ Y) 2 − 2b(X−µx)(Y − µ Y) b 2(X−µ x) 2 Hint Do not expand this expression Keep the terms grouped as they are (a) (4 points) Derive the expected value of U E(U) =µ ′ n = EX n The nth central moment of X is defined as For y > 0, y = −logx implies that x = e−y, ie, g−1(y) = e−y and FY (y) = 1− FX g−1(y) = 1− FX e−y = 1−e−y Hence we may write FY (y) = 1−e−y I(0,∞) This is exponential distribution function for λ = 1 24 CHAPTER 1 ELEMENTS OF PROBABILITY DISTRIBUTION THEORY For continuous rvs we have the followingLµ(µ^;Y) = 0 and that µ^ is a consistent estimator of the unknown value of the parameter, µ0 Weak conditions required for consistency are quite complicated and will not be given here † Taking a Taylor series expansion around µ = µ0 and then evaluating this at µ = µ^ gives 0 ' lµ(µ0;Y)lµµ0(µ0;Y)(µ^¡µ0)
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} v v µ v Ç Z À v µ P } Á Z } ( ð X ð 9 Y }Y v ó X í 9 z }z / v / v v Z µ W Z À v µ /EZ ï î U ò õ ð u o o } v V P } Á Z ï X ó 9 Y }Y v ô X ñ 9 z }zY ¶ µy (1¡µ)n¡y for y 2 f0;;ng Y is binomially distributed with parameters n and µ We write Y » Bin(n;µ) Note that – the number of trials is flxed, – the probability of success is the same for each trial, and – the trials are independent Example Paired Sample Sign Test (contd) Let Y be the number of plants for which the number of lost hours has decreased after theDistributions, since µ and σ determine the shape of the distribution • The rule for a normal density function is e 2 1 f(x;
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Y = n is a binomial distribution with parameters n and λ1 λ1λ2 E(XX Y = n) = λ1n λ1 λ2 3 Consider nm independent trials, each of which results in a success with probability p Compute the expected number of successes in the first n trials given that there are k successes in all Solution Let Y be the number of successes in nm trials Let X be the number of successes in^ u µ v P í ì>W ~Y µ o } u u ^ v P } v ô ï ñ í õ 7L1ILOO D v u µ u 'W lD ÆW l&W ò ô l ð ô l ð î v u U ^ W ( v v P U > > u o µ o ^d/ v o µ v P v P o ( ( µ } v l ~^ t µ v } µ u u Ç P vTitle Microsoft Word Document1 Author Robin Created Date PM
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