}BOBOdHHHHHH H Z $.H.l..... .D.h...1 -ZxHH(FG(HH(dH'@W ՚  0/]@D@;;x/`՚ x/ٝ] @  Th z \DSETT2 (     $g<!!)!1!3wx!  < (C 4D (g 4h ( % 1   " 2   *    3  56 37 5< 3= 5f 3g 5l 3m 5  - 51 2 -3 5n 3o 5t 3u 5x    ' ?;JJOJJEO8_&_& H? cd8 ~>>hH 33x  (, Q( s**- 1 kUU* ? : 9GXV*V*(ٌ H Statistics Examples Mean, Variance, and Standard Deviation Letbe n observations of a random variable X. We wish to measure the average of in some sense. One of the most commonly used statistics is the mean,, defined by the formula  Next, we wish to obtain some measure of the variability of the data. The statistics most often used are the varianceand the standard deviation We have  It is easy to show that the variance is simply the mean squared deviation from the mean. Covariance and Correlation Next, letbe n pairs of values of two random variables X and Y. We wish to measure the degree to which X and Y vary together, as opposed to being independent. The first statistic we will calculate is the covariancegiven by  Actually, a much better measure of correlation can be obtained from the formula  The quantityis known as the coefficient of correlation of X and Y. The Covariance Matrix Covariances and variances are sometimes arranged in a matrix known as a covHariance matrix. In our case, the covariance matrix will be amatrix: ZNDSET.H6H Root EntryF`,ſOle OlePres0002Equation Native `  4 |- .1  & & MathTypePTimes New Roman- 2 `dX 2 `X 2 `X Times New Roman- 2 np Times New Roman- 2 c1p 2 2pTimes New Roman- 2 `,` 2 `M,` 2 `C,`MT Extra- 2 `K-&PMathTypeUUD X 1 ,X 2 ,K,X n & "System-.DX=MEOW X 1 ,X 2 ,K,X nDSET.HDzH/Q#6*V V ??XVHH VVpdXXL@@L|   deh< Fࡱ> Root EntryF@ſOle OlePres0002Equation Native `  4 |- .1  & & MathTypePTimes New Roman- 2 `dX 2 `X 2 `X Times New Roman- 2 np Times New Roman- 2 c1p 2 2pTimes New Roman- 2 `,` 2 `M,` 2 `C,`MT Extra- 2 `K-&PMathTypeUUD X 1 ,X 2 ,K,X n & "System-.DX=MEOW X 1 ,X 2 ,K,X nDSET.H!8f/$6*  ??<HH    ,(  && ࡱ> Root EntryF@ſOle OlePres000Equation Native  4{4  .1  @& & MathTypePSymbol- 2 `@m Times New Roman- 2 AX&,MathTypeUU  m XxJ & "System-.tMEOW m XDSET.H! $]/< $6*  1q q1 ??1qHH 1q1q$0,, $==r[RjfngCCZZ)D<$$$D<H00$["ࡱ> Root EntryF`ſ@Ole OlePres000fEquation Native |  , ; .1  @` &  & MathType "-DD  jSymbol- 2 @m Times New Roman- 2 AX 2  i> 2 i> 2 >npTimes New Roman- 2 X 2 n 2  XSymbol- 2 n= 2 H= Symbol- 2 2={Symbol- 2 Times New Roman- 2 1 Times New Roman- 2 1p;&lMathTypeUU` m X =2X=1nX ii=1n  & "System-.`tMEOW m X =2X=1nX ii=1n DSET.H< J/ $6*   ??HHH  $$@8* ࡱ> Root EntryF ſOle OlePres000Equation Native  8 {{  .1  @@& & MathTypePSymbol- 2 "s Times New Roman- 2 KX Times New Roman- 2 I2p&(MathTypeUU s X2 & "System-.𷂴MEOW s X2DSET.H" </0B $6* nn$J J$ ??($JHH $J$J ( ( 4H^bTsn_9>KK:ࡱ> Root EntryF ſ@Ole OlePres000tEquation Native X : ) .1  @&d & MathType` "-  "-p2-p:dddSymbol- 2 "s 2 s Times New Roman- 2 `KX 2 `XSymbol- 2 x= Times New Roman- 2 T2pTimes New Roman- 2 .`)&HMathTypeUU< s X = s X2 . & "System-.<MEOW s X = s X2 .DSET.H H 1j/` $6* %%H H ??8HHH HH8@H@@`hppxxxx~C;;;88/ࡱ> Root EntryF@_ ſOle OlePres000Equation Native   !z mo .1  &` & MathType@ "-  |  "-r2-r:VVVNSymbol- 2 `"s Times New Roman- 2 KX 2  i> 2 i> 2 3np 2 i> 2  i> 2 <npTimes New Roman- 2 n 2 `X 2  n 2 `XSymbol- 2 `x= 2 ` - 2 k  2   2   2 k 2  2  2  2  2  2 H 2  2  2  2  2 H 2  Symbol- 2 '={ 2 0={Symbol- 2  2  Times New Roman- 2 n1 2 n 1 Times New Roman- 2  2p 2 1p 2 1p 2 f2po&MathTypeUU s X = 1nX i2i=1n  -1nX ii=1n  () 2 {}  & "System-.MEOW s X = 1nX i2i=1n  -1nX ii=1n  () 2 {} DSET.H "/p$6*  ??HH NW`$  WNࡱ> Root EntryF (1ſOle OlePres000Equation Native  {~ _ .1  @& & MathType`Symbol-2 ,(Symbol-2  )Symbol-2 (Symbol-2 * )Symbol-2  (Symbol-2 >)Times New Roman- 2 X 2 Y 2 X 2 Y 2  X 2 Y Times New Roman- 2  np 2 np Times New Roman- 2 1p 2 1p 2 2p 2 2pTimes New Roman- 2 j,` 2 ,` 2 e,` 2  ,` 2  ,` 2 y,`MT Extra- 2  K_&MathTypeUU X 1 ,Y 1 (),X 2 ,Y 2 (),K,X n ,Y n () & "System-.4MEOW X 1 ,Y 1 (),X 2 ,Y 2 (),K,X n ,Y n ()DSET.H "/$6*::  ??\HH  $EH 4"( ࡱ> Root EntryF (1ſOle OlePres000Equation Native  84  .1  &` & MathTypePSymbol- 2 `"s Times New Roman- 2 EXY&0MathTypeUU$ s XY JJ & "System-.4MEOW s XYDSET.H &Ԓ (H)"P* $6* (l(l5 5 ??5HH 554ddddgV&& )E   i.&S  ,,4 &ࡱ> Root EntryFO:ſ Ole OlePres000vEquation Native   !"#~<<  .1   &x & MathType "-``` ` Symbol- 2 "s Times New Roman- 2 EXY 2  i> 2  i> 2 84i> 2 1np 2 i> 2 8 i> 2 1:np 2 i> 2 8i> 2 1!npTimes New Roman- 2 ,n 2 X 2 N Y 2  n 2 X 2 MYSymbol- 2 = 2  - 2   2   2   2  2  2  2  2  2  2 ` 2 ` 2 ` 2 C 2 C 2 ^C 2 $ 2 $ 2 ^$ Symbol- 2 8={ 2 8.={ 2 8={Symbol- 2  2   2 Times New Roman- 2 21 2  1 Times New Roman- 2 81p 2 81p 2 81p&MathTypeUU s XY =1nX i Y ii=1n  -1nX ii=1n  ()Y ii=1n  (){} & "System-.4ǷMEOW s XY =1nX i Y ii=1n  -1nX ii=1n  ()Y ii=1n  (){}DSET.H *w Sx*# $6* UUnI In ??$nIHH nInILLL\\\\\cC:ڗ֖نՈ%pllԚܛ̚Кt$HD8ppl`p (xHHEL|, gZQi)q[[[XX`  Qࡱ>  Root EntryFKſ@Ole OlePres000 Equation Native 80  !"#$%&'()*+,-./012345679:;<=>?@") h   .1   `&  & MathType "-8C8+B B BB "-(Y  "-Y - `` Symbol- 2 @r Times New Roman- 2 -XY 2 ~ i> 2 o i> 2  i> 2  np 2 #i> 2 Ti> 2 np 2 ni> 2 ;i> 2 np 2 i> 2  i> 2 qnp 2 i> 2   i> 2 z np 2 i> 2  vi> 2 np 2 Fi> 2  i> 2 tnpTimes New Roman- 2 x X 2  Y 2 Wn 2 X 2 Y 2 X 2  n 2 X 2 Y 2 n 2 YSymbol- 2 = 2 "- 2 d 2 d 2 d 2  2  2  2 K 2 K 2 K 2  2  2  2  - 2 )  2 )  2 )  2 U 2 U 2 U 2 C 2  2  2  2   2 C 2  2  2  2   2 - 2 # 2 # 2 # 2  2  2  2 C 2  2  2  2   2 C! 2 ! 2 ! 2 ! 2  ! Symbol- 2  ={ 2 ={ 2 ={ 2  e={ 2  n ={ 2  ={ 2  h={Symbol- 2 |  2 ! 2  2  2   2 C 2  Times New Roman- 2  1p 2 $1p 2  1p 2  2p 2  1p 2  1p 2 2p 2 2p 2  F1p 2  1p 2 |2pTimes New Roman- 2 ]1 2 " 1 2 1& MathTypeUU r XY =X i Y ii=1n  -1nX ii=1n  ()Y ii=1n  () X i2i=1n  -1nX ii=1n  () 2 {}Y i2i=1n  -1nY ii=1n  () 2 {}  & "System-.@ȷMEOW r XY =X i Y ii=1n  -1nX ii=1n  ()Y ii=1n  () X i2i=1n  -1nX ii=1n  () 2 {}Y i2i=1n  -1nY ii=1n  () 2 {} DSET.H "' , $6* dd  ??PHH $$?>  ;4/5  ࡱ> Root EntryFKſOle OlePres000Equation Native  84  .1  &@ & MathTypePSymbol- 2 `@r Times New Roman- 2 -XY&(MathTypeUU r XY & "System-.wMEOW r XYDSET.H,-C$6*XX  ??xHH 64<<77++1QT0   ࡱ> Root EntryFRſOle OlePres000Equation Native  0  .1  @&` & MathType Times New Roman- 2 @42 2 @J2Symbol- 2 @6& MathTypeUU 22J & "System-.MEOW 22DSET.H$(T-$6*hh7h h7 ??7hHH 7h7h&$8@444<4DDdld    TM8H8:VFT@444<4DDdldd    hTX\d& R&ࡱ> Root EntryF[ſOle OlePres000Equation Native  L < I .1  @ & z & MathTypeTimes New Roman- 2 4CSymbol- 2 = 2  2 \ 2  2 u  2 \u  2 u Symbol- 2 s 2  s 2 Ts 2 Zs Times New Roman- 2 X 2 -XY 2 ZwXY 2 ZZY Times New Roman- 2 2p 2 N2pI&MathTypeUU| C=s X2 s XY s XY s Y2 [] & "System-.|ĊMEOW C=s X2 s XY s XY s Y2 []DSET.HB 6*DSET.HB6*FNTMCUTSDSUMHDNISTYL2@STYLA@6@n@@@@@ W                                    (   & )  !*)", #$$ %#&('4(! )1!*%!+2!,"!-3$ .5/-!0!1'-HASH%r $ H9;I\J%}#%%& h*h*h- h-!<3 1>1 &>1.'>3 .>3 -2)2'"55 C<1 /C<3 0C>1 +C>3 ,C~<)C~>*D2'(L:Qhjajreqitsr zo}Bodn CHARA   "     @  #    " "  #       #  HASH$ &    C      03 R CELL$A0"HASH  GRPH@A.n.HASHt l RULRxA @@` @L Ab@?@..6 D.B@@.L.u.....@...-.h.q... $.H.l..... .D.h...` $.H.l..... .D.h...@   $.H.l..... .D.h...`@ $.H.l..... .D.h... $.H.l..... .D.h...@ $.H.l..... .D.h...@@  $.H.l..... .D.h...@@  $.H.l..... .D.h...@ $.H.l..... .D.h...HASH  BB [bb  | 6 6 b@@D B6 B6BvBvCDd |6+LKUP 0$(,%#*1& !."//0)+-'.($NAMEDefault Default SSHeaderBodyFooterFootnoteFootnote Index Word4095Null Normal Heading 1 Heading 2Default Paragraph Font List Bullet subscriptList Bullet Last Body TextGreekequationStyle 21Style 22Body Text IndentTitledDFNTMArialArialTimes New RomanCentury SchoolbookSymbolETBL@FNTMCUTSDSUMHDNISTYL ETBL