let x be a rv with e x then 2 xtu2 ie xz my eh x i ell ul
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# Let X be a RV with E [ X ] = . Then: 2Xtu2 ) IE [ XZ - MY ] EH X ' - PowerPoint PPT Presentation

Jul 11, 2023 •17 likes •67 views

Two Games Two Games fair fair Game 1: Flip a coin 10 times. For each Head, Game 2: Flip a coin 10 times. For each Head, you win 100. For each Tail, you lose 100. you win 10000. For each Tail, you lose 10000. Variance Expected Winnings on Flip

  1. Two Games Two Games fair fair Game 1: Flip a coin 10 times. For each Head, Game 2: Flip a coin 10 times. For each Head, you win 100. For each Tail, you lose 100. you win 10000. For each Tail, you lose 10000. Variance Expected Winnings on Flip i : Expected Winnings on Flip i : - - CS 70, Summer 2019 Fi Fi = 100ft ) too ) ( I ) I Effi ] I = Ill I t Effi ] O NOOO ) ) t C- - 0000 = = Lecture 21, 7/30/19 Expected Winnings After 10 Flips: Expected Winnings After 10 Flips: - I - fi , Eff 7=0 Effi O ECF ] ] = - Q: Which game would you rather play? 1 / 26 2 / 26 3 / 26 Definition of Variance Alternate Definition Variance: A Visual The key difference is the variance . We can use linearity of expectation to get an alternate form that is often easier to apply . Variance is the expected “distance” to mean. ' ] IECX uz Var( X ) = - - # Let X be a RV with E [ X ] = µ . Then: 2µXtu2 ) IE [ XZ - MY ] EH X ' I = - Ell - ul - 2µE[ X Var( X ) = x ]tµ2 IECXZ ] linear # = ftp.aais.irenrr - # Tartan always U is . 2 ] = Efx non-neg-ttfustdD-jgxy.TW if - - Value 4 / 26 5 / 26 6 / 26

  2. Variance of a Bernoulli Variance of a Dice Roll Variance of a Dice Roll 25+36 ] f- ( I E [ R 2 ] = Let X ∼ Bernoulli( p ) . What is the variance of a single 6-sided dice roll? 9 16 4 t t t t ÷ .ro Then E [ X ] = p roll { 1,2 , 3,4 , 5,63 R = value dice of a . f- ( 91 ) = What is X 2 ? E [ X 2 ] ? What is R 2 ? - p ) . a = { / ENT tip to - - Ifp RIB - CIEL IE [ R2 ] Wwf X' to Var( R ) = =p * ÷÷÷÷ - EF . af = - FE EXT ) 2 Var[ X ] = ECXZ ] . = F- - P ) - P ) . ) ( Notes ph " Ber ( pp = rare 'D = it Bercp ) Var CX ) X - 7 / 26 8 / 26 9 / 26 Variance of a Geometric Variance of a Geometric II Variance of a Geometric III Recall E [ X ] = 1 Know the variance; proof optional, but good From the distribution of X , we know: p . ① .€ , - p I - - p )2pt practice with manipulating RVs . lP[X=i]=pt( )pth I -_ - Efx 'T - CENT ) ' . . . CX ) - ¥ var - - HEI . ¥ From E [ X ] , we know: Let X ∼ Geometric( p ) . - p )2pt - p -1311 Cfp )p IECX I Strategy: Nice expression for p · E [ X 2 ] 2. - I 1- . . . - - ' I . p - ppp ② I Efx - p ) p 9h t 4h . I t =L t z.pt/4l-plpt6CtpTpt...)tfp-Ci-p7p-ltP5p.ypIEfX4=2IECX = . . . - PIP ] - fit Yf I - - pl Efx 4 C I p t . ② - ① - z sides both Subtract from . - ( - PPP - p - p ) p t 3 I I t 5 C I 1 PIE t . . . = ] 1 - t f- p - ftp.T Efxi for - Li - P2P 2 solve . ) fz.pt/4l-plpt6CtpTpt = ECxy=2 . . 10 / 26 11 / 26 12 / 26

  3. Variance of a Poisson Variance of a Poisson II Break Same: know the variance; proof optional, but Use E [ X ( X − 1 )] to compute Var( X ) . good practice with functions of RVs . - Efx ] ) 2 i ] Efxz ] var ( X ) l deft qf iffy IPCX = = Let X ∼ Poisson( λ ) . → ← - IFECXIJ - - IDTECX ] = Efxlx Strategy: Compute E [ X ( X − 1 )] . Would you rather only wear sweatpants for the . e- X Efx ( X THAY 1 ) ] - = . rest of your life, or never get to wear sweatpants - - E) ! -(# I i I ever again? lastyszu.de X X = = ' Fez E ← taywrreen.es - - - xx e = j 22 =e# I A = 13 / 26 14 / 26 15 / 26 Properties of Variance I: Scale Properties of Variance II: Shift Example: Shift It! Consider the following RV: Let X be a RV, and let c 2 R be a constant. Let X be a RV, and let c 2 R be a constant.  Let E [ X ] = µ . Let E [ X ] = µ . 1 w.p. 0 . 4  Then, let µ 0 = E [ X + c ] =  Utc X = 3 w.p. 0 . 2 Var ( X ) e. Var( cX ) = - FEENY  5 w.p. 0 . 4 varix ) I lin  Var( X + c ) = - 3) C ex ) IE , . Var = Var C X Efffxtc 'T ] ) ) = U varcxtc = - - Cc EXIT What is Var( X ) ? Shift it ! Efe xD = =EAXtEUXY ) - CHE ⇒ it :: : = CHECK ] EXT ) ' * of Vary , - my * ' EAX - = - 0.2=3.2 - 351=4 . 0.8 to IEC X shift . ( X ) * Cz var = ten → - 3 ] = O IEC X 3.2 - 3D Var C C x = 16 / 26 17 / 26 18 / 26

  4. Sum of Independent RVs Variance of a Binomial Sum of Dependent RVs Main strategy: linearity of expectation and Let X 1 , . . . , X n be independent RVs. Then: Let X ∼ Bin( n , p ) . Then, indicator variables Var( X 1 + . . . + X n ) = Var( X 1 ) + . . . + Var( X n ) X = Xitxzt Useful Fact: . . ( X 1 + X 2 + . . . + X n ) 2 = Proof: Tomorrow ! Here, X i ∼ -1 .tl/nBerlp)XiiidTfdVarCX,)tVarCXz)t...tVarlXn)--n.VarCXi Xitxzt Today: Focus on applications. Var( X ) = .tl/nKXitXzt.-.tXn)-=CXftXit...tXn2)tCXiXztXiX3t.-.tXn-iXn . . ) ) Binh ,p ) x - =µpFpTH ) - Y Bincnitp - D - - non - VARY ) - Vari ) Ee Xi - jxi t alter ? = ' , xix ; 19 / 26 20 / 26 21 / 26 HW Mixups (Fixed Points) HW Mixups II HW Mixups III What is S 2 i ? E [ S 2 (In notes.) n students hand in HW. I mix up their i ] ? Using our useful fact : ' ] HW randomly and return it, so that every possible - I : If I E [ S 2 ] = . tsn ) * I Els it t Effs Sat mixup is equally likely . , t si - = Ef En . . - if jsisjl sit Let S = # of students who get their own HW. , linearity student i For i 6 = j , what is S i S j ? E [ S i S j ] ? for : indicator Last time: defined S i = jsisj ] t IEC Effi ? Si 2) HW " i own getting = . . sit :¥÷÷÷÷ Ber Hn ) S i ∼ . Efs - 1) EES , sit s , 2) In t = n n ⇒ T Using linearity of expectation : ÷ . t ECS . ] = Efs Elsie , It tsn I ECS , Sat E [ S ] = t . . . . - = then ) 22 / 26 23 / 26 24 / 26

  5. HW Mixups IV Summary Put it all together to compute Var( X ) . Today: ' - CECXD I Variance measures how far you deviate from Efx 4 C x ) var - = FEW tncn-DEG.SI - mean I Variance is additive for independent RVs ; riff n#n¥n proof to come tomorrow I = t - I Use linearity of expectation and indicator of = variables 25 / 26 26 / 26

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