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Page 75
... stochastic independence of X1 , X2 , ... , Xn Remark . If X1 , X2 , and X , are mutually stochastically independent , they are pairwise stochastically independent ( that is , X , and X ,, i ‡ j , where i , j = 1 , 2 , 3 are stochastically ...
... stochastic independence of X1 , X2 , ... , Xn Remark . If X1 , X2 , and X , are mutually stochastically independent , they are pairwise stochastically independent ( that is , X , and X ,, i ‡ j , where i , j = 1 , 2 , 3 are stochastically ...
Page 77
... stochastically independent and that E ( et ( x1 + x2 ) 1 = - ( 1 − t ) -2 , t < 1 . - 1 2 2.31 . Let X1 , X2 , X3 , and X be four mutually stochastically inde- pendent random variables , each with p.d.f. f ( x ) = 3 ( 1 − x ) 2 , 0 ...
... stochastically independent and that E ( et ( x1 + x2 ) 1 = - ( 1 − t ) -2 , t < 1 . - 1 2 2.31 . Let X1 , X2 , X3 , and X be four mutually stochastically inde- pendent random variables , each with p.d.f. f ( x ) = 3 ( 1 − x ) 2 , 0 ...
Page 134
... mutually stochastically independent . 4.37 . Let X1 , X2 , ... , X , be r mutually stochastically independent gamma variables with parameters a = a , respectively . Show that Y1 = tion with parameters α = α1 + 1 X1 + X2 + + a , and ẞ X ...
... mutually stochastically independent . 4.37 . Let X1 , X2 , ... , X , be r mutually stochastically independent gamma variables with parameters a = a , respectively . Show that Y1 = tion with parameters α = α1 + 1 X1 + X2 + + a , and ẞ X ...
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A₁ A₂ Accordingly best critical region c₁ cent confidence interval chi-square distribution complete sufficient statistic compute conditional p.d.f. confidence interval Consider continuous type critical region decision function defined degrees of freedom denote a random discrete type distribution function distribution having p.d.f. Equation Example EXERCISES function F(x given H₁ hypothesis H independent random variables integral joint p.d.f. k₁ Let X1 Let Y₁ limiting distribution marginal p.d.f. matrix maximum likelihood moment-generating function mutually stochastically independent noncentral normal distribution order statistics p.d.f. of Y₁ P(A₁ p₁ Poisson distribution positive integer probability density functions quadratic form random experiment random interval random sample random variables X1 respectively sample space Show significance level simple hypothesis statistic Y₁ stochastically independent random sufficient statistic theorem unbiased statistic variance o² w₁ X₁ X₁ and X2 X₂ Y₂ Z₁ zero elsewhere μ₁ μ₂ Σ Σ