Introduction to Mathematical StatisticsThe fifth edition of text offers a careful presentation of the probability needed for mathematical statistics and the mathematics of statistical inference. Offering a background for those who wish to go on to study statistical applications or more advanced theory, this text presents a thorough treatment of the mathematics of statistics. |
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... x2 ) X2 £ 211 ( x2x1 ) = E [ u ( X1 , X2 ) ] f ( x1 , x2 ) fi ( x1 ) or u ( x ) f ( x ) dx -∞ 02 = = E [ ( X — μ ) ... [ xx ] - ** n k1X = Σ . Καμία i = 1 n Ni = 1 = Ĉ x2 - X2 1 n ni = 1 war ( ± k , x , ) = Σ kjo ? + 2ΣΣkk , poo , i = 1 i ...
... x2 ) X2 £ 211 ( x2x1 ) = E [ u ( X1 , X2 ) ] f ( x1 , x2 ) fi ( x1 ) or u ( x ) f ( x ) dx -∞ 02 = = E [ ( X — μ ) ... [ xx ] - ** n k1X = Σ . Καμία i = 1 n Ni = 1 = Ĉ x2 - X2 1 n ni = 1 war ( ± k , x , ) = Σ kjo ? + 2ΣΣkk , poo , i = 1 i ...
Page 86
... x2 = 12 X1 Finally , we shall compare the values of We have - X2 2 X2 0 < x2 < 1 . dx Pr ( 0 < X , < ¦ | X2 = 3 ) and Pr ( 0 < X , < ) . Pr ( 0 < xX , < } } x2 = } ) = [ ' " ' " f122 ( x , { i } ) dx , = || X2 3 ) 1/2 - S 1/2 [ ^ ( 3 ) ...
... x2 = 12 X1 Finally , we shall compare the values of We have - X2 2 X2 0 < x2 < 1 . dx Pr ( 0 < X , < ¦ | X2 = 3 ) and Pr ( 0 < X , < ) . Pr ( 0 < xX , < } } x2 = } ) = [ ' " ' " f122 ( x , { i } ) dx , = || X2 3 ) 1/2 - S 1/2 [ ^ ( 3 ) ...
Page 164
... x x = 0 , 1 , 2 , 3 , 3 x ! ( 3x ) ! 3 0 A elsewhere . B ye B , We seek the p.d.f. g ( y ) of the random variable Y = X2 . The transformation y = u ( x ) = x2 maps = { x : x = 0 , 1 , 2 , 3 } onto = { y : y = 0 , 1 , 4 , 9 } . In ...
... x x = 0 , 1 , 2 , 3 , 3 x ! ( 3x ) ! 3 0 A elsewhere . B ye B , We seek the p.d.f. g ( y ) of the random variable Y = X2 . The transformation y = u ( x ) = x2 maps = { x : x = 0 , 1 , 2 , 3 } onto = { y : y = 0 , 1 , 4 , 9 } . In ...
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A₁ A₂ Accordingly approximate best critical region C₁ C₂ chi-square distribution complete sufficient statistic conditional p.d.f. confidence interval Consider continuous type converges in probability correlation coefficient critical region defined degrees of freedom denote a random discrete type distribution function F(x distribution with mean distribution with p.d.f. distribution with parameters dx₁ equation Example Exercise Find the p.d.f. gamma distribution given Hint hypothesis H₁ independent random variables integral joint p.d.f. Let the random Let X1 Let Y₁ limiting distribution marginal p.d.f. matrix moment-generating function order statistics P(C₁ p₁ percent confidence interval Poisson distribution positive integer probability density functions probability set function r₁ random experiment random sample respectively sample space Section Show significance level simple hypothesis subset sufficient statistic t-distribution t₂ theorem unbiased estimator variance o² X₁ X₂ Y₁ Y₂ zero elsewhere μ₁ μ₂ σ²