(1) Central Limit Theorem(2) Order StatisticsMath 370X - Lecture 10Central Limit Theorem and Order StatisticsSaumil PadhyaNovember 14, 2016(1) Central Limit TheoremLet X be a random variable with meanµand standard deviationσ. SupposeX1, X2, ..., Xnare nindependent variables with the same distribution as X.Let Yn= X1+ X2+ ... + Xn. Then E[Yn] = nµ and V ar[Yn] = nσ2.The central limit theoremsays that as n increases,Ynapproaches a normal distribution withmean nµ and variance nσ2.When an exam question asks you to calculate a probability involving sum of a large number ofrandom variables (a general threshold is n≥30), it is asking you to assume that the sum ofrandom variables is normally distributed.For example, an exam question might go like follows - “X follows a binomial distribution withn=50 and p=0.01. What is P(X>30)?”. To find P(X>30), you can apply the binomial formula 20times to get the exact answer but that is not very practical. Thus, the only practical choice is toassume X is normally distributed with the appropriate mean and variance, and use the normaltables to compute P(X>30).(2) Order StatisticsAssumeX1, X2, ..., Xnare independent and have the same distribution. The order statistics ofthese areY1, Y2, ..., Ynin whichY1represents the random variableXiwith the minimum value,and Ynis the random variable Xiwith the maximum value.The density function of the minimum order statisticY1and the maximumYnare easy to computeby applying intuition and the cdf approach -FY1(x) = P [Y1≤ x] = P [min(X1, X2, ..., Xn) ≤ x]FY1(x) = 1 − P [min(X1, ..., Xn) > x] = 1 − P [X1> x] · P [X2> x]...P [Xn> x]FY1(x) = 1 − (1 − FX(x))nAnd you can compute the density fY1(x) by taking the derivative of the cdf.Similarly, we get the distribution function of Ynto beFYn(x) = (FX(x))n1You are rarely told to calculate the density function of an order statisticYk, but here’s the formulafor it -fYk(x) =n!(k − 1)!1!(n − k)![F (x)]k−1[1 − F
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