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 13
... Theorem 2. The probability of the null set is zero ; that is , P ( Ø ) = 0 . Proof . In Theorem 1 , take C = so that C * = 6. Accordingly , we have P ( Ø ) = 1 P ( C ) = 1 − 1 = 0 , and the theorem is proved . — - Theorem 3. If C1 and ...
... Theorem 2. The probability of the null set is zero ; that is , P ( Ø ) = 0 . Proof . In Theorem 1 , take C = so that C * = 6. Accordingly , we have P ( Ø ) = 1 P ( C ) = 1 − 1 = 0 , and the theorem is proved . — - Theorem 3. If C1 and ...
Page 247
... theorem called the central limit theorem . A special case of this theorem asserts the remarkable and important fact that if X1 , X2 , ... , X , denote the observations of a random sample of size n from any distribution having positive ...
... theorem called the central limit theorem . A special case of this theorem asserts the remarkable and important fact that if X1 , X2 , ... , X , denote the observations of a random sample of size n from any distribution having positive ...
Page 255
... Theorem 4 , ( Yn / n ) ( 1 − Y „ / n ) / [ p ( 1 − p ) ] converges in probability to 1 , and Theorem 5 asserts that the following does also : Vn = ( Y / n ) ( 1 — Y / n ) p ( 1 - p ) 1/2 Thus , in accordance with Theorem 6 , the ratio ...
... Theorem 4 , ( Yn / n ) ( 1 − Y „ / n ) / [ p ( 1 − p ) ] converges in probability to 1 , and Theorem 5 asserts that the following does also : Vn = ( Y / n ) ( 1 — Y / n ) p ( 1 - p ) 1/2 Thus , in accordance with Theorem 6 , the ratio ...
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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