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 . Proof . In Theorem 1 , take A = 0 so that A * = A. Accordingly , we have P ( 0 ) = 1 and the theorem is proved . - P ( A ) = 1 1 = 0 , - Theorem 3. If A1 and ...
... Theorem 2. The probability of the null set is zero , that is , P ( 0 ) = 0 . Proof . In Theorem 1 , take A = 0 so that A * = A. Accordingly , we have P ( 0 ) = 1 and the theorem is proved . - P ( A ) = 1 1 = 0 , - Theorem 3. If A1 and ...
Page 196
... 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 items of a random sample of size n from any distribution having finite variance ...
... 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 items of a random sample of size n from any distribution having finite variance ...
Page 224
... theorem , Theorem 2 , p . 213 , n Y1 = † K ( X , ) is a sufficient statistic for the parameter 0. To prove 1 n that Y1 = 1⁄2 K ( X , ) is a sufficient statistic for 0 in the discrete case , 1 1 we take the joint p.d.f. of X1 , X2 ...
... theorem , Theorem 2 , p . 213 , n Y1 = † K ( X , ) is a sufficient statistic for the parameter 0. To prove 1 n that Y1 = 1⁄2 K ( X , ) is a sufficient statistic for 0 in the discrete case , 1 1 we take the joint p.d.f. of X1 , X2 ...
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A₁ A₂ Accordingly c₁ chi-square distribution complete sufficient statistic compute conditional p.d.f. confidence interval 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 hypothesis H₁ independent random variables integral joint p.d.f. k₁ Let the random Let X1 Let Y₁ likelihood ratio limiting distribution marginal p.d.f. moment-generating function mutually stochastically independent noncentral normal distribution order statistics p.d.f. of Y₁ P(A₁ Poisson distribution positive integer probability density functions probability set function 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 t₂ theorem unbiased statistic variance o² W₁ X₁ and X2 X₂ Y₂ Z₁ zero elsewhere μ₁ μ₂ Σ Σ σ²