ALK - Metody Stat. - Wzory 2016 - WZORY Statystyka = = = = = = = = = = = = = = = = = = = = = = = = = = = =(

ALK - Metody Stat. - Wzory 2016 - WZORY Statystyka = = = =...

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Unformatted text preview: WZORY Statystyka ∑ = = = = = = = √ ∑( ̇ − ) = √ ∑( ̇ − ) = = ∧ √ = ∑( − ) − ∧ √ = ∑( ̇ − ) − = ∧ = − ∧ ∧ =√ ∑ = = = √ ∑( − ) ∧ =√ ∑( − ) − ∑ ̇ ∙ = = ∑( − ) =√ = √ ∑ ∙ = = − ∧ = ! = ( ) = (−)! , () = ! ( ∪ ) = () + () , ( ∪ ) = () + () − ( ∩ ) , = ! , ( ∩ ) = () ∙ () , ( / ) = (∩) () , () = − (′ ) ( / ) = (∩) () ( ∩ ) = ( ) ∙ ( / ) , ( ∩ ) = () ∙ ( / ) () = ( ) (/ ) + ( ) (/ ) + ⋯ + ( ) (/ ) = ∑= ( ) (/ ) () = ( ) ∙ ∙ − , ( = ) = , () = ( ≤ ) = ( ( −∞, ]) = µ = ∑ , = = = ∑ − () , = = √ , +∞ () ≥ , ∫−∞ () = , () = ( < ) = ∫−∞ () , ′ () = () ( < ) = ∫−∞ () = () , +∞ ( > ) = ∫ () = − ( < ) = − () , +∞ = ∫−∞ ∙ () − () , ∫ = + ( < < ) = ∫ () = () − () + + , ∶ − (, √) , +∞ = ∫−∞ ∙ () ∫ () = [ () ] = () − () ∫ () = [ () ] = () − () ∶ ( + , √( − ) ) , ( = ) = ( ) ∙ ∙ − , : (, √) , ( = ) = ∙ − , : ( , √ ) 1 ( = ) = − ∙ () = { () = ! ∶ ( , √ ) , , ≤ ≤ − ł √ ∙ − (−µ)2 , ( ) = { ≤ (, ], − = ∙ √ , √ ∶ ( > ∶ (, √ = ∑ = + + ⋯ + + (−) , √ ) ∶ ( , ) , , ( < < ) = () − () , = ∑= , : (, √), = →∞ , − ∶ ( , ) , , ( < ) = () , − = (−) = − () − ⇒ ), , = ~ ( ∙ , √ ) →∞ , = ( + + ⋯ + ) = ∑ ⇒ ~ ( , = →∞ , →∞ , =µ ⇒ ~ ( ∙ , √ ) ⇒ ~ ( , √ Estymator = = = ( ̅ ) = = √ √ − ∧ = √ = ( ̅ ) = = = µ − µ ̂= = = − = ( ̅ ) = ̅ ̂)= = ( ̂ √ + = = ( − ) = = √ ( + ) ̂ − ̂ = Błąd Względny = ( − ) = √ + = √ = − ) ̂̂ ̂)=√ = ( √ ) Błąd Standardowy = ̅ ( − ) + ( − ) + − ̂ ̂ ̂ ̂ ̂ − ̂ ) = √ = ( + = − ̂ − ̂) = ( = = ̂ − ̂ 2 ( − < < + √ = = , ( − = , √ = = √ ̂ ̂ ̂ ̂ )= − , √− , = ( ) = − , = √ )=−, = ̂ ̂ )= − , = + , , √ ̂ ̂ = = ∙ , = , , ( ) = − , = = + = , , ( − ) + ( − ) = √ + − = ∙ √ ( ( − ) − √ + ( ) = − + < − < ( − ) + ∙ √ + ) = − ( − ) + ( − ) , + − ( ) = − , = , , ̂= √ , √− ( ) = − = − , =+ , , = < − < ( − ) + √ + ) = − = √ (( − ) − ∙ √ , ̂− , = ̂+ , , = = ∙ , = = − , = + , ̂ + ∙ √ < < ( ( − ) − √ + = < < + ∙ = = , ̂ − ∙ √ ( = − , =+ = = , ) = − , ( ) = − √ < < + √− ( − ∙ + , = ∙ , = < − < ( − ) + √ + ) = − = √ + , = = ∙ , = , 3 , ̂ − ̂) ( ( − ∙ √ ̂ = ̂ = , ̂ ̂ ̂ ̂ ̂ ̂ ̂ ̂ ̂ − ̂) + < − < ( + )= − + ∙ √ , ( ) = − ∗ = − = − = ̂− = √ √ , ( ) = − ∙ √ ̂= − √ = = ( ) = − , , ( ) = − = = ∙ , , ̂ , − , ̂ − ̂ ) , , , ̂ ̂ − = √ − , ( , ) ( ) = − , ( ) = − , , ( , ) ∙ ( + ) ( ) = − ( ) = − , + ̂ − ̂ √ ∙ = , , ∑ + = + , , ∙√ + + + − = , ( ) = − − √ ( ) = − , ̂ ̂ + = = ( ) = − , = √ , (w.est.: √ , − = , .. ( − ) ∑= ∑= = √ , = , − ∙ + , ∑ ( − ) ( ) = − , ( ) = − ( − ) ∑= ∑= , = = ( ∑( − ) ) ∙ ( ∑( − ) ) = ∑= ∙ = ∑= ∙ = ∑= ∑= , ∙ = ∙ =√ , ∙ = ∙ , ∙ ((−),(−)) = ∙ ∙ , , = √ + = ∙ , ∶ (−)∙(−) , 4 = ∑= ( − ) = ∑= , ̅ ̅ (, ) = ∑ − ∑ = − , ( − ) ∶ −− , ∑ − ̅ ̅ = , = − , = ̂ = + ( − ) , ∈ , = = (, ) √ − , √ − = √ − √ − ∑ − ̅ ̅ ∑ −( ̅ ) { = ̅ − ̅ = ̂ = √ ∑( − ∑( ) ̂ ) √ = − − , = √ , = √ ∑ ∙ ∑( − ̅ ) ∑ ∑( − ̅) = + + [ ] [ ] [ ] = + ( − ∙ < < + ∙ ) = − ( − ∙ < < + ∙ ) = − = , − = − , = − − = = = , = ∑( − ) ∙ = ∑ ∑( − ) , = 5 ∑ ( − ) = ∑ ( ̂ − ) + ∑ ( − ̂ ) = = = zmienność SS MS (stopnie swobody) /sum of squares/ /mean squares/ ∑ (̂ − ) − CAŁKOWITA ∑ ( − ̂ ) − Statystyka testowa − = Odchyleń + Regresji = = − − = − ∑ ( − ) = 6 ...
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