# notes09 - STATISTICS 330 Mathematical Statistics...

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STATISTICS 330 Mathematical Statistics Supplementary Lecture Notes Cyntha A. Struthers Dept. of Statistics and Actuarial Science University of Waterloo Waterloo, Ontario, Canada Fall 2009

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Contents 1P R E V I EW 1 1 .1 Examp l e............................. 1 1 .2 Examp l 3 2R A N D OMV A R I A B L E S 5 2 .1 In t rodu c t ion. .......................... 5 2 .2 D i s c r e t eRandomV a r iab l e s .................. 6 2 .3 Con t inuou sRandomV a r l e s................. 8 2 .4 F un c t ion so faRandomV a r l e ............... 10 2 .5 Exp e c ta t 13 2 .6 M om en tG e ra t ingF c t s ................ 17 2 .7 Lo ca t ionandS c a l eP a ram e t e r 19 2 .8 Ca l cu lu sR ev i ew . ....................... 20 3 Joint Distributions 23 3 .1 Jo in tandM a rg ina lCDF s................... 23 3 .2 Jo tD i s c r e t a r l e s ............... 24 3 .3 Jo tCon t a r l e s ............. 25 3 .4 Ind ep end tRandomV a r l e s................ 29 3 .5 Cond i t iona lD i s t r ibu t s ................... 31 3 .6 Jo tExp e c t s ....................... 35 3 .7 Cond i t lExp e c t ................... 37 3 .8 Jo tM e t c t 39 3 .9 B iv a r ia t eNo rm a i s t r t ................ 40 4 Functions of Random Variables 43 4 t c t 43 4 .2 On e - t o -On eB a r t eT ran s fo a t s............ 44 4 .3 M e t c t ionM e thod . ........... 47 1
0 CONTENTS 5 Limiting or Asymptotic Distributions 51 5 .1 Conv e rg en c einD i s t r ibu t ion. ................. 51 5.2 Convergence in Probability . 53 5 .3 L im i tTh eo r em s......................... 56 6 Estimation 61 6 .1 In t rodu c t .......................... 61 6.2 Method of Maximum Likelihood -On eP a ram e t e rCa s e ..................... 62 6.3 Method of Maximum Likelihood -Mu l t ipa e t e s e..................... 66 6.4 Asymptotic Properties of M.L. E s t ima t o r s-On a e t e s e............... 73 6 .5 In t e rv a lE s t a to r s....................... 74 6 .6 R e la t iv eL ik e l ihood. ...................... 78 6.7 Asymptotic Properties of M.L. E s t t o r s-Mu l t e t e s e .............. 80 6.8 Con f d c eR eg ion 83 7H y p o t h e s i sT e s t s 8 9 7 t c t 89 7.2 Likelihood Ratio Tests for Simple Hypo th e s e s ........................... 90 7.3 Likelihood Ratio Tests for Composite e s e 93

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Chapter 1 PREVIEW The follow examples will illustrate the ideas and concepts we will study in STAT 330. 1.1 Example Thefo l low ingtab leg ive sthenumbe ro ffumb le sinagamemadeby110 Division A football teams during one weekend: No. of Fumbles: x 0 1 2 3 4 5 6 7 8 Total Obs. Frequency: f x 8 24 27 20 17 10 3 1 0 110 It is believed that a Poisson model will f tthesedatawe l l .Whym ight this be a reasonable assumption? (PROBABILITITY MODELS) If we let the random variable X = number of fumbles in a game and assume that the Poisson model is reasonable then the probability function (p.f.) of X is given by P ( X = x )= μ x e μ x ! x =0 , 1 ,... where μ is a parameter of the model which represents the mean number of fumbles in a game. (RANDOM VARIABLES, PROBABILITY FUNC- TIONS, EXPECTATION, MODEL PARAMETERS) Since μ is unknown we might estimate it using the sample mean ¯ x = 8(0) + 24(1) + ··· +1(7) 110 = 281 110 2 . 55 . (POINT ESTIMATION) The estimate ˆ μ x is the maximum likelihood (M.L.) estimate of μ . It is the value of μ which maximizes the likelihood 1
2 CHAPTER 1. PREVIEW function. (MAXIMUM LIKELIHOOD ESTIMATION) The likelihood function is the probability of the observed data as a function of the un- known parameter(s) in the model. The M.L. estimate is thus the value of μ which maximizes the probability of the observed data.

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## This note was uploaded on 10/21/2010 for the course MATHEMATIC stats taught by Professor Various during the Spring '10 term at Waterloo.

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notes09 - STATISTICS 330 Mathematical Statistics...

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