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Wzory statystyka

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Wzory statystyka

  1. 1. STATYSTYKA OPISOWA Analiza struktury ∑= = n i ix n x 1 1 ∑= ⋅= k i ii nx n x 1 1 ∑= ⋅= k i ii nx n x 1 ˆ 1 ∑= = n i i H x n x 1 1 ∑= = n i i i H x a n x 1 ∑= = n i i i H x a n x 1 ˆ N nx x k j jj og ∑= ⋅ = 1 ( ) N nxx N nxS S k j jj k j jj og ∑∑ == − += 1 2 1 2 2 )( ( )∑= −= n i i xx n xS 1 22 1 )( ( ) i k i i nxx n xS ⋅−= ∑=1 22 1 )( ( ) i k i i nxx n xS ⋅−= ∑=1 22 ˆ 1 )( %100 )( )( ⋅= x xS xV )()( 2 xSxS = ∑= −= n i i xx n xd 1 1 )( i n i i nxx n xd ⋅−= ∑=1 1 )( i n i i nxx n xd ⋅−= ∑=1 ˆ 1 )( )()( xSxxxSx typ +<<− { } { }ii xxR minmax −= ( )∑= −= n i i xx n xM 1 3 3 1 )( ( ) i k i i nxx n xM ⋅−= ∑=1 3 3 1 )( ( ) i k i i nxx n xM ⋅−= ∑=1 3 3 ˆ 1 )( )( )( 3 3 3 xS xM =γ )(xS Dx As − = ( ) ( ) D DDDD DD D x nnnn nn xD ∆⋅ −+− − += +− − 11 1 β β β ββ Q Q i x n ncumn xQ ∆⋅ −⋅ += − )( 1 2 13 QQ Q − = %100⋅= Me Q VQ )()( )()( 13 13 QMeMeQ QMeMeQ AQ −+− −−− = Analiza korelacji ( )( ) ( ) ( )∑ ∑ ∑ = = = −⋅− −− = ⋅ = n i n i ii n i ii xy yyxx yyxx ySxS yx r 1 1 22 1 )()( ),cov( r ( ) ( )1 6 1 2 1 2 −⋅ −⋅ −= ∑= nn rr R t i yx S %10022 ⋅= xyrR ( ) ∑∑= = − = r i k j ij ijij n nn 1 1 2 2 ˆ ˆ χ , n nn n ji ij •• ⋅ =ˆ { }1,1min 2 −−⋅ = krn V χ 2 2 χ χ + = n C ( )( )11 2 −− = krn T χ )()()(( 21211122211211 21122211 nnnnnnn nnnn +++ − =ϕ baxy +=ˆ 2 11 2 1 11 2 )( ),cov( )( )(       − − === ∑∑ ∑ ∑∑ == = == n i i n i i n i n i i n i iii xy xxn yxyxn xS yx xS yS ra xayb −= dcyx +=ˆ 2 11 2 1 11 2 )( ),cov( )( )(       − − === ∑∑ ∑ ∑∑ == = == n i i n i i n i n i i n i iii xy yyn yxyxn yS yx yS xS rc ycxd −=
  2. 2. ( ) 22 ˆ )( 1 2 1 2 2 − = − − = ∑∑ == n u n yy uS n i i n i ii ( ) ( ) )( )()( ˆ 2 2 1 2 1 2 2 yS uS n kn yy yy n i ii n i ii ⋅ − = − − = ∑ ∑ = = ϕ 22 1 ϕ−=R Analiza dynamiki ctct yyd −=/ 1/ −−= ctct yyd c ct ct y yy − =∆ / 1 1 1/ − − − − =∆ t tt tt y yy c t ct y y i =/ 1 1/ − − = t t tt y y i 1 1 1 1/ 1 1/2/31/2 ... −−− − ==⋅⋅⋅= n nn n n nng y y iiiii ∑ ∑ ∑ ∑ = = = = ⋅ ⋅ === n i ii n i itit n i i n i it t w qp qp w w w w I 1 00 1 1 0 1 0 F q F p L q P p P q L pw IIIIIII ⋅=⋅=⋅= ∑ ∑ ∑ ∑ = = = = = ⋅ ⋅ = n i n i p n i ii n i iit L p w iw qp qp I 1 0 1 0 1 00 1 0 ∑ ∑ ∑ ∑ = = = = = ⋅ ⋅ = n i p t n i t n i iti n i itit P p i w w qp qp I 1 1 1 0 1 P p L p F p III ⋅= ∑ ∑ ∑ ∑ = = = = = ⋅ ⋅ = n i n i p t n i ii n i iti L q w i w qp qp I 1 0 1 1 00 1 0 ∑ ∑ ∑ ∑ = = = = = ⋅ ⋅ = n i p n i t n i iit n i itit P q iw w qp qp I 1 0 1 1 0 1 P q L q F q III ⋅= Trend baty +=ˆ ( )∑ ∑ ∑∑ ∑ = = == = −⋅ ⋅−⋅ = n t n t n t t n t n t t ttn yttyn a 1 1 22 11 1 tayb −= ( ) 22 ˆ )( 1 2 1 2 2 − = − − = ∑∑ == n u n yy uS n i t n t tt ( ) ( )∑ ∑ = = − − = n t tt n t tt yy yy 1 2 1 2 2 ˆ ϕ 22 1 ϕ−=R ( ) ( )∑= − − ++⋅= n t p tt tT n uSyS 1 2 2 1 1)()( ( )∑= −= in i tt i i yy n O 1 ˆ 1 0 1 =∑= i d i O ∑= = in i t t i i y y n S 1 ˆ 1 dS i d i =∑=1 )ˆ( ittt Oyyz +−= ittt Syyz ⋅−= ˆ ip ObaTy ++= )( ip SbaTy ⋅+= )( ( ) ( )∑= − − ++⋅= n t i tp tt tT n zSyS 1 2 2 1 1)()( %100 ˆ )( ⋅= p p w y yS b 2 )( 1 2 2 − = ∑= n z zS n t t t
  3. 3. Rachunek prawdopodobieństwa )()( xXPxF <= ∑< = xx i i pxF )( ∫∞− = x dttfxF )()( ∑= ⋅= n i ii pxXE 1 )( ∫ ∞ ∞− ⋅= dxxfxXE )()( [ ]∑= ⋅−= n i ii pXExXD 1 22 )()( [ ]∫ ∞ ∞− ⋅−= dxxfXExXD )()()( 22 [ ]22 )()( XEXEXD −= ( ) [ ]222 )()( XEXEXD −= ∑= ⋅= n i ii pxXE 1 22 )( ∫ ∞ ∞− ⋅= dxxfxXE )()( 22 )()( 2 XDXD = ccE =)( 0)(2 =cD )()( XEcXcE ⋅=⋅ )()( 222 XDcXcD ⋅=⋅ )()()( YEXEYXE +=+ )()()( 222 YDXDYXD +=+ dla zmiennych niezaleŜnych )()()( YEXEYXE −=− )()()( 222 YDXDYXD +=− dla zmiennych niezaleŜnych )()()( YEXEXYE ⋅= dla zmiennych niezaleŜnych )()(2)(2)()()( 222 YEXEXYEYDXDYXD −++=+ )()( xXPXF <= )()()( aFbFbXaP −=≤≤ ∑<≤ =≤≤ bxa i i pbXaP )( ∫=≤≤ b a dxxfbXaP )()( σ mX Z − =    − == p p xXP 1 )( 0 1 = = x x pXE =)( )1()(2 ppXD −= knk qp k n kXP −       == )( ),1( pq −= nk ,...,2,1,0= npXE =)( npqXD =)(2 ! )( k e kXP k λ λ − == pn ⋅=λ , ,...2,1,0=k λ=)(XE λ=)(2 XD Estymacja przedziałowa α σσ αα −=       +<<− 1 n uxm n uxP ααα −=       − +<< − − 1 11 n S txm n S txP ααα −=       +<<− 1 n S uxm n S uxP α χ σ χ αα −=           << −−− 12 1, 2 1 2 2 2 1, 2 2 nn nSnS P ασ αα −=       +<<− 1 22 n S uS n S uSP αρ αα −=       − +<< − − 1 11 22 n r ur n r urP ααα −=             − +<< − − 1 )1()1( n n m n m u n m p n n m n m u n m P 2 22 d u n σα ≥ 2 22 d Su n α ≥ 2 2 )1( d ppu n − ≥ α 2 2 4d u n α ≥
  4. 4. Testy parametryczne 00 : mmH = 00 : mmH ≠       < > 00 00 : : mmH mmH n mx u σ 0− = −σ znane 10 − − = n S mx t ,30≤n 1−n -stopni swobody n S mx u 0− = 30>n 2 0 2 0 : σσ =H 2 0 2 0 : σσ >H 2 0 2 2 σ χ nS = ,30≤n 1−n stopni swobody 122 2 −−= ku χ 1−= nk 30>n 00 : ppH = 00 : ppH ≠       < > 00 00 : : ppH ppH n pp p n m u )1( 00 0 − − = 210 : mmH = 210 : mmH ≠       < > 210 210 : : mmH mmH 2 2 2 1 2 1 21 nn xx u σσ + − = −21,σσ znane 2 2 2 1 2 1 21 n S n S xx u + − = 12021 ≥+ nn       + −+ + − = 2121 2 22 2 11 21 11 2 nnnn SnSn xx t 12021 <+ nn 221 −+ nn stopni swobody 2 2 2 10 : σσ =H 2 2 2 10 : σσ >H 2 2 2 1 ˆ ˆ S S F = 2,1 21 −− nn - stopni swobody 210 : ppH = 210 : ppH ≠       < > 210 210 : : ppH ppH n qp n m n m u 2 2 1 1 − = , gdzie: 21 21 nn mm p + + = , pq −=1 , 21 21 nn nn n + = 0:0 =ρH 0:0 ≠ρH       > < 0: 0: 0 0 ρ ρ H H 2 1 2 − − = n r r t 122≤n n r r u 2 1− = 122>n Testy nieparametryczne ( ) ∑= − = r i i ii np npn 1 2 2 χ )(xS xx u G iG i − = 1−− rk stopni swobody )1()( )2(2 1 2 21 2 21 212121 21 21 −++ −− − + − = nnnn nnnnnn nn nn k U ( ) ∑∑= = − = r i t j ij ijij e en 1 1 2 2 χ n nn e ji ij •• ⋅ = , )1)(1( −− kr stopni swobody

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