variance and covariance July 23,2020 f Applications of of Expectation Linearity Variance 2 Covariance and correlation 3 Last time Distributions p if Xsl Bernoulli CP x PREM ipso p i f pin pj I Xu PHX Bincngp Linearity of Expectation e E Exif ECC Cnxn EWE Xi
1 Applications of Linearity of Expectation Hand back assignments Example n students to at random back their hw Expected number of students who get Xn the number of students who get back tn their hw ttsIhnedirn tbEs gmaE r.r.fi use indicator others 0 IT In PrEIj Xn I In e EE In E EE Ide c In Xn Iie E ECTS ETI Mtn linearity I E a Prodi lxprEIEDeoxprcq.ro a E I s achoo't I e n The expectednumber of students who get their own a class size he is home works in
me balls Throw Example in bins n Be The expected number of bins empty X number of empty bins Pr Help Indicators help m ith bin is empty n 2 Etsi otherwis P resist B n linearity EI Xs t Ist n EEN EEE Ii EE.CIIS.EE 5senInT.ED is Eaxprosi ajslxprczisD DXPRET.es E IITs I m aC 0,12 n s m EG E n her and For n h hut l 1a C EX 37N hero n of 37 tinamou n1 Ie will bins be empty
2J Variance.se Random Walk Examle Particle 4 p p r r B N J X a I N p p g r 3 S 4 4 0 5 6 Z l I 3 G z a The Particle dimension in moves one to the right or a At each Zime Step Particle moves with equal P.robability left the Particle after The Position of Sn n moves PRED E I the 2th movement Xi L pre'T Z n Eti O X Sn Xn Xze 1 E n E j Xi EEN o C Csn gootuseful CDxprcxz.se 73 t.xprexi D IaxPrcai E xD C 1 Nz s 1 12 D AGH R the expecteddistance from 0 Isnt what is 0
work SnI we ttiiiiii.axn5 E.xiezEjxixoEES.n Xie Xn C EE.fi Ee2Ey.xixi3 2 Eg.Efxixi3 Ei E ECxE3e n C lfprcxz.si lfprcxi D EGxprfdisaJ E xE 1 12 aef 1 12 119 I a Xjsb a bxPr XI E C EXiXj3s a betel a xPrCXj b Xi pr m pr Xist PVCxjsD 1 4 I I PrCXj preexist 53 sq prcxis DPrcxjsD 4 GD fyc yprcxis.BPrfl.jo D variance of Sn 1 14 1 1,2 0 late 1k
Definition with expectation Ecosef For X r v a the variance of X is defined to be w T Varys E Ux what does variance measure The deviation from value mean FarTX The square root is Cx the standard deviation variable X of EE h J EExFEEEff Var Theoreme Ms X e5 proof E Vaux linearity K2 2 1 V2 E ECE ECD EM 12 EE N 2N r E Ext v2 Some properties of Variance Var Varun Xtc exercise I Varexy exercise Var CX i
More Examples Uniform distribution a uniform random variable Xa the set 91 non on Uniformly off PIKE In VIELE zu E Ex ECE Var X EEN E.fi n5a a3 aEa 9 n InaEa tnnengDEEX23 h NII and Ea2 PrExsa a aef's TM 2 4 42 In 2 44 DR s 6 h Varcx I 2
Hand back assignments Example n students to random at Xn the number of students who get back their hw pr student i gets their hw yn ft Use indicators f I otherwise In Es Ii exercise Ans Ii e Iz i ECM Ty Hitt Eis ZEE Var X in E EXT's E EXT I Var X EEN t zig Eui Ij fi ECI ECCE Iit EUR3 Ii Pr Ii Of Ln l Pr ZED on DE cz a s2xPrkisDxPIEf on.Tjen napairEEXT of indecis th 2 t 2 E L Z haha heh 1 varCX s2
Independent Random Variables For independent random variables Theorem X Y have we EIN ELY E CHIT independent Proof of Eae.AE gbaPrExsagY b 2 X and Y EEEYI ab Prasad PREYS aaPrEx a3JxfbEbPrcxsbD sEEXJrEEYT.D For independent random variables Theorem y we have Varchevarly varcxey proof C Expect'D Elway 53 f Exey Varulay EEx5ezE EEY7eEEyI E 212 linearity GO.EE ig zEqyyypfqy2J EExI 2EexLEEYJ7EY7 CEEx9 E Ext e ENT Eey5 t2 EEXY7ECx3ECyD
Vav.LK eVaVCy EExyJ ECx3ECyT l2 are independent o and y If x Vary star Varcxey independent then x and y are If remember Varun Evan Varun y Varix Var Atty Harty t e Dwarcy var CA Varix Vary heads Example is the number of in kn n tosses a biased coin with heads probability P of Bin en p ithtoss In xn Eti A fine p n n Kris EEE Ii3s hip E P E EE Io E Is jet inn
E PHP Ishii EI var s Var Varun Ii n PhD f 2 EE zig C Ii 23 Var Ii PE p p's Pel p ExpresisBeozptiso covariance and correlation 3 of random variables Definition The covariance X and Y is CoV XM EEK G LY k Ds E Ext EEN E p f Remarks If Xgy are independent CoV X Y O L we say Can if Roy are independent corex y so No of iz true not converse is the E EXT ECD s E Ex XJ 2 Xix Cov C EXT ECB rate x
bilinear is 3 Covariance For any collection of random variables gym and fixed constants fy In A big skim Then ai a san g EaibjCovLxisyj aixigo.ES bjYjJs Cov 4 CoV CR y About determines how X and Y The sign 1 X y of related are Is hard The magnitude to interpret 2 Definition Correlation suppose X and Y are random variables so The o and with CY correlation GCN of x and y is corrcx.yjscorcx.yjo.CN y
Theorem I feorrcxgyjflX ayebifaso ocorrcx.gs 1 if correngyys l a t Corre E l Corry A s EExIEaD ecgyg.gg gsECxY3 corrciy 647601 Ysaxsb E xcaxebD ECx3EEaxeb lat X Gex 6caxebjsevaraf.bg 64 aEEXD ibEXD aEEXJ.BE Exsa2Varcx M Lal Vara t Ei i a a fairy ASO l
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