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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Page 89
... x Xx ( x3 ) ) dx ] f ( x ) dx , = = = [ * -∞ < -∞ X2 fi ( x1 ) E ( X2 \ x1 ) ƒ1 ( x1 ) dx1 E [ E ( X2 | X1 ) ] , which is the first result . Consider next , with μ2 = E ( X2 ) , var ( X2 ) = E [ ( X2 — μ2 ) 2 ] = - - = E { [ X2 — E ( X2 | ...
... x Xx ( x3 ) ) dx ] f ( x ) dx , = = = [ * -∞ < -∞ X2 fi ( x1 ) E ( X2 \ x1 ) ƒ1 ( x1 ) dx1 E [ E ( X2 | X1 ) ] , which is the first result . Consider next , with μ2 = E ( X2 ) , var ( X2 ) = E [ ( X2 — μ2 ) 2 ] = - - = E { [ X2 — E ( X2 | ...
Page 407
... X ? / 0 ' has a chi - square distribution with n n n degrees of freedom . Since α = PrΣ X2 / 0 ′ ≥ c / 0 ′ ; H。 c / 0 ' may be read from Table II in Appendix B and c determined . Then C = n { ( x , xx ) : x ≥ c } i X2 , Xn 1 is a ...
... X ? / 0 ' has a chi - square distribution with n n n degrees of freedom . Since α = PrΣ X2 / 0 ′ ≥ c / 0 ′ ; H。 c / 0 ' may be read from Table II in Appendix B and c determined . Then C = n { ( x , xx ) : x ≥ c } i X2 , Xn 1 is a ...
Page 451
... X2 , ... , X , be a random sample from a normal distribution N ( u , 2 ) . Show that n n n 1 Σ ( x ; − x ) 2 = Σ ... Xx . ... .. ) 3 . i = 1 j = 1 k = 1 - i = 1 j = 1 k = 1 i = 1 = Show that -- i = 1 j = 1 k Sec . 10.1 ] The ...
... X2 , ... , X , be a random sample from a normal distribution N ( u , 2 ) . Show that n n n 1 Σ ( x ; − x ) 2 = Σ ... Xx . ... .. ) 3 . i = 1 j = 1 k = 1 - i = 1 j = 1 k = 1 i = 1 = Show that -- i = 1 j = 1 k Sec . 10.1 ] The ...
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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. g₁(y₁ gamma distribution given Hint hypothesis H₁ independent random variables integral Let the random Let X1 Let Y₁ limiting distribution marginal p.d.f. matrix moment-generating function order statistics p.d.f. of Y₁ P(C₁ p₁ Poisson distribution positive integer probability density functions probability set function R₁ r₂ random experiment random sample respectively sample space Section Show significance level simple hypothesis subset sufficient statistic t-distribution t₂ theorem unbiased estimator variance o² W₁ X₁ X₂ Y₁ Y₂ zero elsewhere μ₁ Σ Σ σ²