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# INTRO STAT THEORY I STAT 702

GPA 3.93

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This 68 page Class Notes was uploaded by Shane Marks on Monday October 26, 2015. The Class Notes belongs to STAT 702 at University of South Carolina - Columbia taught by Staff in Fall. Since its upload, it has received 13 views. For similar materials see /class/229677/stat-702-university-of-south-carolina-columbia in Statistics at University of South Carolina - Columbia.

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STAT 702J702 December 2nd 2004 Lecfure 28 lnstructor Brian Habing Department of Statistics Telephone 8037773578 Email habingstatscedu V4 STAT 7021702 BHabing Univ ofSC Today Some Notes on the Exam The Results from Last Time 0 Examples Course Evaluations w STAT 7021702 BHabing Univ ofSC Distributions o If ZN01 then 22 92de 0 If X1 Xk are independent xzwith df v1 vk then ZXi xzdfivi STAT 7021702 BHabing Univ ofSC w o If ZNO1 and X 98de are independent then Z X v dev 39 U defm and V xzdfn independent then UmVn Fd fmn V4 STAT702J702 BHabing UnivofSC Relationshigs If X1 Xn are iid NW9 then quot 152 N12 02 dfn71 X I SJ dfn71 V4 STAT702J702 BHabing UnivofSC If X1 an are iid mepxz and Y1 Yny are iid NuYGY2 then 2 s 2 OX N F 2 dfnX 71nY 71 SY STAT 7021702 BHabing Univ ofSC w Example 1 A random sample of battery lifespans of size 20 shows an average of 154 hours and standard deviation of 28 hours Is this enough evidence to reject the manufacturer s claim that the batteries have an average lifespan of at least 16 hours STAT 702J702 BHabing Univ ofSC Example 2 What is the distribution of the difference of two means Example 3 Let W1 wk be iid NuWo12 x1 xL be iid meef Y1 YM be iid NuY622 Z1 zN be iid Nuz622 How can we see if of of STAT 702J702 BHabing Univ ofSC w STAT 702J702 October 21st 2004 Lecfure 78 lnstructor Brian Habing Department of Statistics Telephone 8037773578 Email habingstatscedu V4 STAT 7021702 BHabing Univ ofSC Today Convolutions and Quotients cont Order Statistics STAT 7021702 BHabing Univ ofSC w 361 Special Case 1 Convolution In general say ZXY We can find a general formula for FZZPZltZ simply by finding the appropriate area under fxy Taking the derivative then gives us the pdf STAT 7021702 BHabing Univ ofSC w Example X and Y are exponential RVs with parameter 9 STAT 702J702 BHabing Univ ofSC w 361 Special Case 2 Quotient A general formula for the quotient ZYX can also be derived by examining the CDF To do this easily note that if yxsz then if xgt0 we have y g xz and if xlt0 then y 2 xz STAT 702J702 BHabing Univ ofSC w Back to the earlier example X and Y have joint pdf fXYXy 2 Osxltyg1 UXY STAT 702J702 BHabing Univ ofSC w 37 Order Statistics Let X1 X2 Xn be independent random variables with the same CDF FXX The values in order from lowest to smallest are the order statistics X1 X2 STAT 702J702 BHabing Univ ofSC w First consider the maximum UXn Note that Usu if and only if all of the Xi su FUu PU S u PX1 SunnXn 21 PX1SuPXn Eu STAT 702J702 BHabing Univ ofSC w PX1 uPXn Eu FXu39 FXu FXun Taking the derivative we get fU an FX n 1 STAT 702J702 BHabing Univ ofSC w The minimum VX1 works similarly FVV 11 FX V fVV an V1 FX VHH l STAT 702J702 BHabing Univ ofSC w w Similar logic helps to get the marginal pdf for any of the order statistics as you will show in the homework n me 0 00 39Fk 1xk1 FX xkn k me xk STAT 702J702 BHabing Univ ofSC w n Example 1 Say you conduct 10 independent tests of hypotheses How small should the smallest p value be for you to reject it at a 005 level That is what is the 5thile for the 1St order statistic STAT 702J702 BHabing Univ ofSC V4 Example 2 x00 for a uniform random variable This is a beta distribution with parameters k and n k1 STAT 702J702 BHabing Univ ofSC M 13 a a m 37 a g i i i g I i i I DU DZ 04 05 DE WU DU 02 04 DE D8 in I l m 1 v Q 2 a 7 37 34 c E39 I i i i I I C I I i i I I UEI U2 D4 DE DE TU EIEI U2 U4 DE DE TD I v Avg STAT702J702 BHabing UnivofSC 14 l V D V 3 m HI I a N D 3 of f 00 O2 O4 06 08 10 V4 STAT 702J702 BHabing Univ ofSC The joint pdf of all of the order statistics is nfx1 fxn V STAT702J702 BHabing UnivofSC 16 One way to find the joint pdf of a pair of order statistics would be to integrate out the n2 you are not concerned with Another way is to use what the text calls a differential argument Theorem A on 101 uses this to prove the result in the hmwk STAT 702J702 BHabing Univ ofSC 2 g 17 Say we want the joint pdf of X0 and X0 where iltj The trick to getting the joint pdf directly is try to let our insights into discrete distributions apply to continuous random variables In particular we will imagine that fxy Px XSxclxySY ydy fx PxSX Sxdx V m STAT 702J702 BHabing Univ ofSC 2 g 18 And so fXlXj xixj n 39FXl71xi FX xm FX x0 Jgtze1 1 FX x 11 39fX xo fX 051 V 19 STAT 7021702 BHabing Univ ofSC 2 g STAT 702J702 November 14th 2006 L ecz ure 22 Instructor Brian Habing Department of Statistics Telephone 8037773578 Email habingstatscedu STAT 702702 BHabing Univ of SC 4 Today Last Time Covariance Moment Generating Functions Applications STAT 702702 BHabing Univ of SC 4 Covariance COVX YEXiux Yiuy STAT 702702 BHabing Univ of SC 4 Correlation COVX Y p CorX Y XY JVarXVarY Aw STAT 702J702 BHabing UniVOfSC 45 Moment Generating Functions The momentgenerating function mgf ofX is Mz Ee X M f 2 etme Ma OieiX xwx STAT 702702 BHabing Univ of SC Why moment generating Assume the mgf exists on some interval around 0 Ma DietXfOch M39m i 7einltxgtdx dt 00 STAT 702702 BHabing Univ of SC 4 Other properties a The mgf uniquely determines the pdf b If Ya bX then MY2 ea MXbz C If X and Y are independent and ZXY then MZ2 MX2 MY2 STAT 702702 BHabing Univ 0fSC Example 1 XUniform01 M Xt M aX b ea STAT 702J702 BHabing UniV of SO m Example 2 Sum of Negative Binomials petY MU 1 1 may fort lt 1111 p STAT 702702 BHabing Univ 0fSC Application 3 Intelligent Searching and Sampling a Group Testing A large number n of blood samples are to be tested for a relatively rare disease Can we find all the infected samples in fewer than ntests ltltltlt STAT 702702 BHabing Univ of SC nu 10 M rPVNQ E Consider 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 V M STAT 702702 BHabing Univ of SC n 11 ltlt E Now consider that we divide the 177 samples into mgroups of size k each STAT 702J702 BHabing Univ of SC 12 STAT 702J702 November 11th 2004 Lecfure 23 lnstructor Brian Habing Department of Statistics Telephone 8037773578 Email habingstatscedu V4 STAT 7021702 BHabing Univ ofSC Today 0 Application 2 Intelligent Searches and Sampling 0 Application 3 Random Sums STAT 7021702 BHabing Univ ofSC w Intelligent Searching and Sampling 3 Group Testing A large number n of blood samples are to be tested for a relatively rare disease Can we find all the infected samples in fewer than ntests STAT 7021702 BHabing Univ ofSC w Consider 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 702J702 BHabing Univ ofSC w Now consider that we divide the m samples into mgroups of size k each STAT 702J702 BHabing Univ ofSC w b Stratified Sampling Imagine that a population is natually divided into ngroups or strata What happens if you randomly sample from each stratum separately than it is to take a single random sampling STAT 702J702 BHabing Univ ofSC How can we get an unbiased estimate of the population mean based on the separate strata means STAT 702J702 BHabing Univ ofSC w What is the variance of imam STAT 702J702 BHabing Univ ofSC w When is stratified sampling better STAT 702J702 BHabing Univ ofSC w Random Sums An insurance company receives N independent claims XNin a given time period Where Nis also a random variable independent of the Xi What are the mean and variance of N T 2 2X1 i1 STAT 702J702 BHabing Univ ofSC w w This would be much easier to work with if we could condition on Nand consider 739V ETNnEXi Nnj i1 my 1391 1509 nEltXgt 1391 STAT 702J702 BHabing Univ ofSC w n But we somehow need to take the expectation over N as well STAT 702J702 BHabing Univ ofSC V4 The general result is EYEXEYXYX A similar result is VarYVarEYXEVarYX w STAT 7021702 BHabing Univ ofSC 13 M Chapter 4 Revisited More on Expected Values Recall that 1M 2 mm 2 loxfxdx VarX 2x mzpm o 2 x2fxdx V mm STAT 702702 BHab1n Un1V 0fSC g quotmint For constants a and b Ea bX a b EX Vara b X b 2 VarX STAT 702702 BHab1n Un1V 0fSC M g quotmint Let X1 X2 XIn be mutually independent random variables then zx EZiXi ZiEXi ZXi GZXZ VarZiXi ZiVarXi 2 0X3 STAT 702702 BHab1n Un1V 0fSC M g quotmint What if the Xi are not independent First if the Xi have joint pdf fX1Xn and YgX1Xn then EY iigx1mxnfx1xndx1dxn Provided the integral converges with lg I STAT 702702 BHabing Univ of SC 4 Now consider Y abZXi i1 and finding EY and VarY STAT 702702 BHabing Univ ofSC Covariance COVX YEX1ux YHY Correlation or 2 XY VarXVaIY STAT 702J702 BHabing UniV ofsC m 45 Moment Generating Functions The momentgenerating function mgf ofX is Mz Ee X M f 2 etme Ma OietXfltxgtdx STAT 702702 BHabin Univ of SC g m3an 10 Why moment generating Assume the mgf exists on some interval around 0 STAT 702702 BHabing Univ of SC r M 11 Other properties a The mgf uniquely determines the pdf b If Ya bX then MY2 ea MXbz C If X and Y are independent and ZXY then MZ2 MX2 MY2 STAT 702702 BHabin Univ of SC g m3an 12 STAT 702J702 November 30th 2004 Lecfure 27 Instructor Brian Habing Department of Statistics Telephone 8037773578 Email habingstatscedu V4 STAT 702J702 BHabing Univ ofSC Today 0 An important Lemma 0 x2 distribution and the variance 39 fdistribution and the mean 0 Fdistribution and variances w STAT 702J702 BHabing Univ ofSC Lemma le1 Xn are iid NW9 then X and X1 X2 Xn 4 are independent STAT 702J702 BHabing Univ ofSC w 12 distribution and the variance The 98de is a special case of the gamma distribution where or V2 and 212 and HS the degrees of freedom STAT 702J702 BHabing Univ ofSC Recall that If ZN01 then Z2 xzd If X1 Xk are independent xzwith df V1 Vk xzdfzvi w STAT 702J702 BHabing Univ ofSC Theorem If X1 Xn are iid n 1s2 Nu62 then T N ijin w STAT 702J702 BHabing Univ ofSC Definition If ZN01 and X 92de are independent then Z Z N d v X f V STAT702J702 BHabing UnivofSC Theorem If X1 Xn are iid X yNt SJ dfn71 NW9 then STAT 702J702 BHabing Univ ofSC w Definition If U xzdfm and V xzdfm independent then UmVn Fd fmn Theorem If X1 an are iid mepxz and Y1 Yny are iid Nmmf then 2 O39 X NFdfnX71nY71 2 Y STAT 702J702 BHabing Univ ofSC 23 o STAT 702J702 Augusts 29th 2006 Instructor Brian Habing Department of Statistics LeConte 203 Telephone 8037773578 Email habingstatscedu M STAT 702702 BHabing Univ of SC 3 Today Sections 11 13 Sample Spaces amp Probability Sections 1516 Conditional Probability amp Independence M STAT 702702 BHabing Univ of SC 3 Chapter 1 Probability M y Webster s Ne W World Dictionary defines probability as 1 a being probable likelihood 2 something probable M STAT 702702 BHabing Univ of SC 3 Looking up probable doesn t really seem to apply likely to occur or be STAT 702702 BHabing Univ of SC is a g5 So we could check likelihood a being likely to happen probability Clearly the dictionary won t be too helpful here STAT 702702 BHabing Univ of SC is a g5 One way of defining the probability of an event is The probability of an event is the proportion of times relative frequency that the event is expected to occur when an experiment is repeated a large number of times under identical conditions ltltltlt STAT 702702 BHabing Univ ofSC null lm E This is the frequentist idea that we will use to connect our theoretical probability constructions to reality Our goal now is to set up a definitional foundation to allow us to apply this intuition in practice STAT 702702 BHabing Univ of SC 4 Experiment A procedure or process that produces results that cannot be predicted with certainty A fair coin is fIppeo z Wce STAT 702702 BHabing Univ of SC 4 Sample Space The set S of all the possible outcomes sample points 0 of the experiment HH HT TH TT Sample spaces can be either discrete or con nuous STAT 702702 BHabing Univ of SC 4 We would get the probability of the events A and B simply by adding the probabilities assigned to their sample points A Exactly one Head HTTH H BF irst ip was a Head HHHT Hm STAT 702702 BHab1n Unlv of SC M g quotanimal 13 Fundamental Types of Events Complement not AC Null Event CD Intersection and n A n B ltIgt gt disjoint Union or inclusive U STAT 702702 BHab1n Unlv of SC M g quotanimal 14 Properties of Probability obtained from the three axioms A PAC 1 PA PCD O C If ACB then PA s PB Proof Since BAUBnAC PBPAPBnAC2PA D Addition Law PAUB PA PB PAnB W STAT 702702 BHab1n Unlv of SC M g quotanimal 15 Conditonal Probability For two events A and B with PBgtO Define the probability of A given B as PAB PAnB PB STAT 702702 BHabin Univ of SC g an film 16 4Exacz y one Head HTTH Frsz 2740 was a Head HHHT PAB PAnB PB PHTPB 12 12 STAT 702702 BHabing Univ of SC 393 17 Multiplication Rule PAnB PAB PB or PBA PA STAT 702702 BHabing Univ of SC 393 18 Independence Definition Two events A and B are independent PAlBPA and PBAPB STAT 702702 BHabin Univ of SC g min Film 19 a Note that if PBgtO then PAlBPA implies that PBAPB by the multiplication rule PAlBPA 3 PA nB PABPB PAPB PBAPA 3 PBAPB STAT 702702 BHabin Univ of SC g min Film 20 a Definition A1 An are mutually independent events if and only if for every suboolleotion A Aik of size k2 n PAi1nAi2 n nAik PAH PAiZ quotPAik STAT 702702 BHabin Univ of SC g min Film 21 a STAT 702J702 October 24th 2006 Lecz ure 7 7 lnstructor Brian Habing Department of Statistics Telephone 8037773578 Email habingstatscedu STAT 702702 BHabing Univ ofSC Today Functions of Jointly Distributed Random Variables Order Statistics STAT 702702 BHabing Univ of SC 4 For the continuous case the joint pdf of Ug1X Y and VgZX Y is fUVU V ampYh1ui V h2ua V where 71 and 72 are the inverse functions xh1u v yh2u v a du dv And J IS the Jacobian J a Avg du dv STAT 702702 BHabing Univ ofSC m Example 2 X and Y are bivariate normal with means 0 variances 1 and correlation O Let r x2 y2 and 9 tan391 Z x Find the joint and marginal distributions STAT 702702 BHabing Univ of SC 4 361 Special Case 1 Convolution In general say ZXY We can find a general formula for FZZPZltZ simply by finding the appropriate area under fXy Taking the derivative then gives us the pdf STAT 702702 BHabin Univ of SC g mr lm a Example X and Y are exponential RVs with parameter A STAT 702702 BHabin Univ of SC g mr lm a 361 Special Case 2 Quotient A general formula for the quotient ZYX can also be derived by examining the CDF To do this easily note that if yxsz then if xgtO we have y s xz and if XltO then y 2 xz STAT 702702 BHabing Univ of SC 4 Back to the earlier example X and Y have joint pdf fxmy 2 0 S xlt yS1 UXY STAT 702702 BHabing Univ ofSC 37 Order Statistics Let X1 X2 XIn be independent random variables with the same CDF FXX The values in order from lowest to smallest are the order statistics STAT 702702 BHabing Univ ofSC First consider the maximum UXn Note that Usu if and only if all of the Xi Su FU u PU S u PX1 u HXn 14 PX1 uPXn SL1 V S A 02 02 b fSC I T T7 J7 BHa in UniV0 g at 10 PX1 guy1009 Su Xu FXMFXMquot Taking the derivative we get fUW nf X UHFX MW 1 V S A 02 02 b fSC I T T7 J7 BHa in UniV0 g at U The minimum VXm works similarly FVV 11FXVn f V V an V1 FAVEquot 1 V S A 02 02 b fSC 1 T T7 J7 BHa in UniV0 g at 12 Similar logic helps to get the marginal pdf for any of the order statistics as you will show in the homework n 3 k1nk fXxk Fk 1Xk1 F X Mk nk rm STAT 702702 BHabin Univ of SC g M13an 13 Example 1 Say you conduct 10 independent tests of hypotheses How small should the smallest p value be for you to reject it at a 005 level That is what is the 5thile for the 1St order statistic STAT 702702 BHabing Univ ofSC r M 14 STAT 702J702 October 5th 2006 L ecz ure 7 3 Instructor Brian Habing Department of Statistics Telephone 8037773578 Email habingstatscedu STAT 702702 BHabing Univ ofSC Today Functions of Continuous Distributions continued Joint Distributions STAT 702702 BHabing Univ of SC 4 Yg X Let X be a continuous RV with pdf fX and Yg X where g is differentiable and strictly monotone every where that fXgtO Then 1 d 1 fyyfXg y g y dy STAT 702702 BHabing Univ ofSC Page 62 For any specific problem it is usually easier to proceed from scratch than to decipher the notation and apply the proposition STAT 702702 BHabing Univ of SC is a g5 Example Let XZ2 where ZNO1 STAT 702702 BHabing Univ of SC is a g5 Example 2 Let XF391U where U is uniform on 01 and F is a CDF ltltltlt STAT 702702 BHabing Univ ofSC rrnil l ltm E Chapter 3 Joint Distributions The joint behavior of two random variables X and Y is determined by there CDF FXYX J PXSX YSJ STAT 702702 BHabing Univ of SC 4 We can use this definition to find the area of any given rectangle PX1lt X3 X2 Y1lt Y3 Y2 FXYX2a Y2 FXYX1a Y2 FXYX2a Y1 quot39 FXYX1a Y1 for x1ltx2 y1lty2 STAT 702702 BHabing Univ ofSC 32 Discrete RV s For discrete RV s the joint pmf is P y PXX YJ STAT 702702 BHabing Univ ofSC Example A fair coin is tossed three times Let Xnumber of heads in three tossings and Y difference in absolute values between the number of heads and number of tails STAT 702J702 b fS C r BHa in Univ 0 g nui lun 10 The Marginal pmf ofX is pxx Zy pX y The Conditional pmf ofX is poXI y PXXIYJ PXX Yy PYJ PXYC J PYJ STAT 702J702 BHab1n Un1V of SC M g uni fl lun 11 X and Y are independent if FXYXa Y FXX FX V This implies that pXYXa Y pxX pYJ It also works for functions gx and hy STAT 702J702 BHab1n Un1V of SC M g uni fl lun 12 33 Continuous RV s Continuous X Y have joint odf FXYXYPXSX Y5 The joint pdf is fXYXaJ 82 FXYXa Y axay V PM K39 nui flint 13 STAT 702702 BHabing Univ of SC So F x y yifuvdudv and NOW 6 A iii f x ydxdy V b f MooX STAT 702702 BHa in Univ 0 SC STAT 702J702 Augusts 24th 2004 Instructor Brian Habing Department of Statistics LeConte 203 Telephone 8037773578 Email habingstatscedu STAT 7021702 BHabing Univ ofSC Today Sections 12 13 Sample Spaces amp Probability Sections 1516 Conditional Probability amp Independence STAT 7021702 BHabing Univ ofSC w Chapter1 Probability continued One way of defining the probability of an event is The probability of an event is the proportion of times relative frequency that the event is expected to occur when an experiment is repeated a large number of times under identical conditions STAT 7021702 BHabing Univ ofSC w This is the frequentist idea that we will use to connect our theoretical probability constructions to reality STAT 702J702 BHabing Univ ofSC w Experiment A procedure or process that produces results that cannot be predicted with certainty A air coin is pped twice STAT 702J702 BHabing Univ ofSC w Sample Space The set Q of all the possible outcomes sample points 0 of the experiment HH HT TH TT Sample spaces can be either discrete or continuous STAT 702J702 BHabing Univ ofSC w An event usually denoted by capital letters at the beginning of the alphabet is a set of sample points AExacf0 one Head HTTH Frsz 17 was a Head HHHT We will discuss several particular kinds of events later w STAT 702J702 BHabing Univ ofSC A probability measure on Q is a function P from events subsets of Q to the real numbers 0 to 1 satisfying the following three axioms 1 PQ 1 2 PA 2 0 for any event A c9 3 If A1 and A2 disjoint then PAiu A2 PAi PAz IfA1 An are mutually disjoint then PA1UAzU UAnU Z11 toooPAi w STAT 702J702 BHabing Univ ofSC The coin ipping example is discrete so we can define the probability measure by giving a probability to each of the sample points so that the sum is l PHHPHTPTHPTTO25 w STAT 702J702 BHabing Univ ofSC We would get the probability of the events A and B simply by adding the probabilities assigned to their sample points AExactly one Head HTTH PA 1A 1A 12 BFirst ip was a Head HHHT PB 1A 1A 12 STAT 702J702 BHabing Univ ofSC a w Fundamental Types of Events Comglement not AC Null Event lntersection and n A n B lt1 gt disjoint Union or inclusive u V4 STAT702J702 BHabing UnivofSC 11 Progerties of Probability obtained from the three axioms A PAC 1 PA P 0 C If ACB then PA s PB M Since BAuBnAC PBPAPBnAC2PA D Addition Law PAUB PA PB PAoB W STAT 702J702 BHabing Univ ofSC V4 Conditonal Probability For two events A and B with PBgt0 Define the probability ofA given B as PAB P AmB PB V4 STAT702J702 BHabing Univ ofSC 13 AExacf0 one Head HTTH Frsz 17 was a Head HHHT PAB PAmS PB PHTPB 2 2 STAT 702J702 BHabing Univ ofSC w M Multiglication Rule PAnB PAB PB or PBlA PA STAT 702J702 BHabing Univ ofSC V4 Independence Definition Two events A and B are independent if PABPA and PBlAPB STAT 702J702 BHabing Univ ofSC w w Note that if PBgt0 then PABPA implies that PBlAPB by the multiplication rule PABPA 3 PA nB PABPB PAPB PBlAPA 2 PBlAPB STAT 702J702 BHabing Univ ofSC w 17 Definition A An are mutually independent events if and only if for every subcollection AH Aik of size k2 n PAi l AiZ 7 nAik PAii PAiZ quot39PAik STAT 702J702 BHabing Univ ofSC V4 STAT 702J702 September 26st 2006 L ecz ure 7 0 lnstructor Brian Habing Department of Statistics Telephone 8037773578 Email habingstatscedu STAT 702702 BHabing Univ ofSC Today Homework Solutions Continuous Random Variables STAT 702702 BHabing Univ of SC 4 H7 Wisconsin has approx 4000000 registered voters of which 4 percent are undecided for the upcoming election a Why is it ok to model a survey of 100 of these voters as a binomial rather than a hypergeometric b In a random sample of 100 registered voters what is the probability of having no undecided respondents STAT 702702 BHabing Univ of SC 4 c How many do you expect to have to survey before you have the first undecided respondent d What is the probability that the tenth person you talk to is your second undecided STAT 702702 BHabing Univ of SC 4 22 Continuous Variables Consider a random number generator that selects a real number at random from between 0 and 1 STAT 702702 BHabing Univ ofSC The probability density function pdf fX satisfies the following a fX 2 O for all X b iofxdx 1 C PaltXltbbjfxdx STAT 702702 BHabing Univ ofSC The Cdf is defined the same way FX Psz ifxdx M STAT 702702 BHabing Univ 0fSC For expected values we need to Change the summation into an integral 1M 2 mm 2 ixfltxgtdx VarX 2x 2px 2 x 2fxdx STAT 702702 BHabing Univ 0fSC Three notes 39 fXF X Pa ltXltbFbFa PXltXlt 0 z fX0 X STAT 702702 BHabing Univ 0fSC Example Consider the distribution with the pdf shown below V S A 02 02 b fSC i T T7 J7 BHa in UniVo g M13an 10 221 Exponential Distribution Consider a Poisson process with parameter 9L Say we are interested in the random variable Wtime until the next occurence STAT 702702 BHabing Univ of 80 r M 11 Lets look at the problem in terms of the Cdf FW PW S W 1 PW gtW 1 Pno Changes in 0 W STAT 702702 BHabing Univ ofSC r M 12 We can then find the mean variance STAT 702J702 BHabing Univ ofSC m 13 quot Now back to our earlier example Between 2am and 4am cars pass the mile marker at a rate of 24 per houn How long until the next car passes V M STAT 702702 BHabing Univ 0fSC M 14 ltlt E 222 The Gamma Distribution 106 t He M fort 2 0 N05 W STAT 702702 BHabing Univ 0fSC 15 STAT 702J702 October 26th 2006 L ecz ure 7 8 lnstructor Brian Habing Department of Statistics Telephone 8037773578 Email habingstatscedu STAT 702702 BHabing Univ ofSC 4 Today Order Statistics STAT 702702 BHabing Univ of SC 4 37 Order Statistics Let X1 X2 XIn be independent random variables with the same CDF FXX The values in order from lowest to smallest are the order statistics STAT 702702 BHabing Univ of SC 4 First consider the maximum UXn Note that Usu if and only if all of the Xi Su FU u PU S u PX1 u HXn 14 PX1 uPXn SL1 V S A 02 02 b fSC I T T7 J7 BHa in UniV0 g at 4 PX1 guy1009 Su Xu FXMFXMquot Taking the derivative we get fUW nf X UHFX MW 1 V S A 02 02 b fSC I T T7 J7 BHa in UniV0 g at s The minimum VXm works similarly FVV 11FXVn f V V an V1 FAVEquot 1 V S A 02 02 b fSC 1 T T7 J7 BHa in UniV0 g at 6 This method is a bit messier to use for the other order statistics Another option is the differential argument The trick to getting the joint pdf directly is try to let our insights into discrete distributions apply to continuous random variables STAT 702J702 BHabing UniVOfSC And we get n fXkXk k1nk fXXk Fk 1Xk1 F X Mk nk STAT 702702 BHabing Univ ofSC Example 1 Say you conduct 10 independent tests of hypotheses How small should the smallest p value be for you to reject it at a 005 level That is what is the 5thile for the 1St order statistic STAT 702702 BHabing Univ ofSC Example 2 xltkgt for a uniform random variable This is a beta distribution with parameters k and nk1 Nerf STAT 702702 BHabing Univ of SC mm n 10 quot quot391 L17 397 C E39 C El r 4 4 Ln EC III III LC II 239 I I I I I I 39239 I I I I I I Ill I12 ELI1 DE DE LEI Ill 32 34 DE DE LEI t 1 CI 392 m W Ln 3947 Equot E39 C LEI Ln III III S I I I I I I I I I I I I ELI I12 Il r DE DE LEI Ill I12 04 DE DE LEI t Nerf STAT 702702 BHabing Univ of SC mm n 11 quot qr LIT 0quot Cr m I LIT 11 N 4 C3 04 L11 l I 30 32 34 36 38 10 12 ASKf STAT 702702 BHabing Univ of SC mm n 12 E The joint pdf of all of the order statistics is x0 39 39 nfx1 39fxn STAT 702702 BHabing Univ of SC 393 13 One way to find the joint pdf of a pair of order statistics would be to integrate out the 172 you are not concerned with Another way is to use what the differential argument Say we want the joint pdf of X0 and X0 where iltj STAT 702702 BHabing Univ of SC 14 And 30 fXlXj x0 9 xj n 39ltz 1gto z 1gtltn jgt 39FXi1Xi X 3 F X X0 ji1 1 FX x0 j 39fX XifX 3cm STAT 702702 BHabing Univ of SC m3an 15 STAT 702J702 November 16th 2006 L ecz ure 23 lnstructor Brian Habing Department of Statistics Telephone 8037773578 Email habingstatscedu STAT 702702 BHabing Univ of SC 4 Today Applications STAT 702702 BHabing Univ of SC 4 Application 3 Intelligent Searching and Sampling a Group Testing A large number n of blood samples are to be tested for a relatively rare disease Can we find all the infected samples in fewer than ntests STAT 702702 BHabing Univ of SC 4 Consider 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 702702 BHabing Univ of SC is a g5 Now consider that we divide the 177 samples into mgroups of size k each ltltltlt M STAT 702702 BHabing Univ ofSC n E Application 4 Stratified Sampling Imagine that a population is naturally divided into n groups or strata What happens if you randomly sample from each stratum separately than it is to take a single random sampling STAT 702702 BHabing Univ of SC How can we get an unbiased estimate of the population mean based on the separate strata means STAT 702702 BHabing Univ 0fSC a g5 What is the variance of 375mm STAT 702702 BHabing Univ 0fSC a g5 When is stratified sampling better ltltltlt STAT 702702 BHabing Univ 0fSC M 9 an Film E Application 5 Random Sums An insurance company receives N independent claims XN in a given time period Where Nis also a random variable independent of the X What are the mean and variance of N TZL i1 ltltltlt STAT 702702 BHabing Univ of SC nu 10 M P WN E This would be much easier to work with if we could condition on Nand consider 739N N ETNnE XilNznj i1 STAT 702702 BHabing Univ of SC mm mm 11 But we somehow need to take the expectation over N as well K STAT 702702 BHabing Univ of SC 12 Chebyshev s Inequality relates the probability of being within a certain range of the mean to the variance for any distribution STAT 702702 BHabin Univ 0fSC g an film 16 Chebyshev s Inequality PX gt k0 lfxdx s l 39X 39fltxgtdx x ugtk039 x ugtk039 0 s l x 22fltxgtdx 2 x ugtk039 k 0 STAT 702702 BHabin Univ 0fSC g an film 17 s O HO fxdx 2 2 00 k 039 1 00 2 2 2 ie m fxdx k 0 oo 2 039 1 2 2 2 k 039 k V STAT 702702 BHabing Univ 0fSC 18 So PX ugtk7si k2 STAT 702702 BHabing Univ of SC is a g5 19 Weak Law of Large Numbers Sect 52 Chebyshev s inequality can be used to prove the weak law of large numbers If X1Xi is a sequence of independent random variables with mean u and variance 2 then Plnlgt8 gt0 as n gtoO STAT 702702 BHabing Univ of SC is a 20 g5 This is an example of Convergence in Probability The probability of being far away from the limit goes to zero as n gtoo ltltltlt M STAT 702702 BHabing Univ ofSC mm 21 E Unfortunately the LLN doesn t give us a good feeling for how close should be to p The central limit theorem provides some guidance in this respect The most common version of the CLT saysthat STAT 702702 BHabing Univ of SC 393 22 Let X1 X2 be a sequence of independent identically distributed random variables with mean u and variance 02 Then n Z Xi hm P 21 s x CIgtx We 7 K J 00 lt X lt 00 Q i STAT 702J702 BHabing Univ ofSC 23 STAT 702J702 November 9th 2004 Lecfure 22 lnstructor Brian Habing Department of Statistics Telephone 8037773578 Email habingstatscedu V4 STAT 702J702 BHabing Univ ofSC Today 39 Moment Generating Functions cont 0 Application 1 System Failure 0 Application 2 Search Methods w STAT 702J702 BHabing Univ ofSC 45 Moment Generating Functions The momentgenerating function mgf of X is MUEe X MU ZefXPOC Mt TetXfxdx w STAT 702J702 BHabing Univ ofSC M Properties of mgf s a If the mgf exists on an interval around zero then Mk0EXk bThe mgf uniquely determines the pdf c If Ya bX then MY1 ea MXbz df X and Y are independent and ZXY then MZ1 MX1 MYU STAT702J702 BHabing UnivofSC Example 1 XUniform01 Mxt Mamba STAT 702J702 BHabing Univ ofSC w Example 2 Sum of Negative Binomials pe Y Mt l1petr fortlt 1n1 p STAT 702J702 BHabing Univ ofSC w System Failures Examples a Suppose a system consists of n identical independent components connected in a series it fails if any one of the components fails What is the probability that the system will fail if each has failure probability p STAT 702J702 BHabing Univ ofSC w b What is the probability the following circuit fails assuming each component is independent with failure probability p in 77 39l 39 STAT 702J702 BHabing Univ ofSC w 8 0 Each component in the system below has an independent exponentially distributed lifetime with parameter 9 Find the cdf and density of the system s lifetime jgi 7 STAT 702J702 BHabing Univ ofSC Intelligent Searching A large number n of blood samples are to be tested for a relatively rare disease Can we find all the infected samples in fewer than n tests V STAT702J702 BHabing UnivofSC 10

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