1STAT 702/J702 B.Habing Univ. of S.C. 1STAT 702/J702 November 16th, 2006-Lecture 23-Instructor: Brian HabingDepartment of StatisticsTelephone: 803-777-3578E-mail: [email protected] 702/J702 B.Habing Univ. of S.C. 2Today • Applications!STAT 702/J702 B.Habing Univ. of S.C. 3Application 3:Intelligent Searching and Samplinga) Group Testing: A large number nof blood samples are to be tested for a relatively rare disease. Can we find all the infected samples in fewer than ntests?2STAT 702/J702 B.Habing Univ. of S.C. 4Consider the case of splitting each of nsamples in half. Combine half of each one is placed into a large combined pool.Should this work better?STAT 702/J702 B.Habing Univ. of S.C. 5Now consider that we divide the msamples into mgroups of size keach…STAT 702/J702 B.Habing Univ. of S.C. 6Application 4: Stratified SamplingImagine that a population is naturally divided into ngroups or strata. What happens if you randomly sample from each stratum separately than it is to take a single random sampling?3STAT 702/J702 B.Habing Univ. of S.C. 7How can we get an unbiased estimate of the population mean based on the separate strata means?STAT 702/J702 B.Habing Univ. of S.C. 8What is the variance of ?strataySTAT 702/J702 B.Habing Univ. of S.C. 9When is stratified sampling better?4STAT 702/J702 B.Habing Univ. of S.C. 10Application 5: Random SumsAn insurance company receives Nindependent claims X1,…. XNin a given time period. Where Nis also a random variable (independent of the Xi ).What are the mean and variance of ∑==NiiXT1STAT 702/J702 B.Habing Univ. of S.C. 11This would be much easier to work with if we could condition on Nand consider T |N.⎟⎠⎞⎜⎝⎛∑====NiinNXEnNTE1|)|(⎟⎠⎞⎜⎝⎛∑==niiXE1∑==niiXE1)()(XnE=STAT 702/J702 B.Habing Univ. of S.C. 12But we somehow need to take the expectation over N as well.5STAT 702/J702 B.Habing Univ. of S.C. 13The general result isE(Y)=EX[EY|X(Y|X)]STAT 702/J702 B.Habing Univ. of S.C. 14A similar result isVar(Y)=Var[E(Y|X)]+E[Var(Y|X)]STAT 702/J702 B.Habing Univ. of S.C. 15Application 6: Interpreting VariancesIf a population is normal (or nearly normal) then the variance is fairly easy to interpret.What if the population is not normal?6STAT 702/J702 B.Habing Univ. of S.C. 16Chebyshev’s Inequality relates the probability of being within a certain range of the mean to the variance for any distribution.STAT 702/J702 B.Habing Univ. of S.C. 17Chebyshev’s Inequality=>− )|(|σμkXP∫=>−σμkxdxxf||)(∫−≤>−σμσμkxdxxfkx||222)()(∫−≤>−σμσμkxdxxfkx||)(||STAT 702/J702 B.Habing Univ. of S.C. 18∫−≤∞∞−dxxfkx)()(222σμ∫−=∞∞−dxxfxk)()(1222μσ22221kk==σσ7STAT 702/J702 B.Habing Univ. of S.C. 19So: 21)|(|kkXP ≤>−σμSTAT 702/J702 B.Habing Univ. of S.C. 20Weak Law of Large Numbers (Sect. 5.2)Chebyshev’s inequality can be used to prove the weak law of large numbers:If X1,….Xi,… is a sequence of independent random variables with mean μ and variance σ2, thenas n →∞.0)|(| →>−εμnXPSTAT 702/J702 B.Habing Univ. of S.C. 21This is an example of Convergence in Probability. The probability of being far away from the limit goes to zero as n→∞.8STAT 702/J702 B.Habing Univ. of S.C. 22Unfortunately the LLN doesn’t give us a good feeling for how close should be to μ.The central limit theorem provides some guidance in this respect. The most common version of the CLT says that:XSTAT 702/J702 B.Habing Univ. of S.C. 23Let X1, X2, … be a sequence of independent identically distributed random variables with mean μ and variance σ2. Then