Introduction to Mathematical Statistics |
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Page 12
... Theorem 2. The probability of the null set is zero , that is , P ( 0 ) = 0 . 0 so that A * = A. Accordingly , Proof . In Theorem 1 , take A = we have P ( 0 ) 1 = - = P ( A ) 1 - 1 = 0 , and the theorem is proved . Theorem 3. If A , and ...
... Theorem 2. The probability of the null set is zero , that is , P ( 0 ) = 0 . 0 so that A * = A. Accordingly , Proof . In Theorem 1 , take A = we have P ( 0 ) 1 = - = P ( A ) 1 - 1 = 0 , and the theorem is proved . Theorem 3. If A , and ...
Page 224
... theorem , Theorem 2 , p . 213 , Y1 = K ( X , ) is a sufficient statistic for the parameter 0. To prove Σ n 1 n that Y1 = 1⁄2 , K ( X , ) is a sufficient statistic for ℗ in the discrete case , 1 we take the joint p.d.f. of X1 , X2 ...
... theorem , Theorem 2 , p . 213 , Y1 = K ( X , ) is a sufficient statistic for the parameter 0. To prove Σ n 1 n that Y1 = 1⁄2 , K ( X , ) is a sufficient statistic for ℗ in the discrete case , 1 we take the joint p.d.f. of X1 , X2 ...
Page 355
... theorem is complete . Theorem 2. Let Q1 and Q2 denote random variables which are quadratic forms in the items of a random sample of size n from a distribu- tion which is n ( 0 , o2 ) . Let A and B denote respectively the real symmetric ...
... theorem is complete . Theorem 2. Let Q1 and Q2 denote random variables which are quadratic forms in the items of a random sample of size n from a distribu- tion which is n ( 0 , o2 ) . Let A and B denote respectively the real symmetric ...
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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