Introduction to Mathematical Statistics |
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Page 309
... degrees of freedom n and 1 , respectively . In Section 8.9 we were able to show that n ( X μ ) 2 and nS2 are stochasti- cally independent . This stochastic independence immediately implies ( Section 4.7 , p . 145 ) that nS2 / 02 has a ...
... degrees of freedom n and 1 , respectively . In Section 8.9 we were able to show that n ( X μ ) 2 and nS2 are stochasti- cally independent . This stochastic independence immediately implies ( Section 4.7 , p . 145 ) that nS2 / 02 has a ...
Page 312
... degrees of freedom . Now a Q2 = b 2 ( X ,. – X ) 2 ≥ 0. In accordance with the theorem , Q1 and Q2 are Σ i = 1 - - 2 stochastically independent , and Q2 / 02 has a chi - square distribution with 1 degrees of freedom . ab 1 . - - - a ...
... degrees of freedom . Now a Q2 = b 2 ( X ,. – X ) 2 ≥ 0. In accordance with the theorem , Q1 and Q2 are Σ i = 1 - - 2 stochastically independent , and Q2 / 02 has a chi - square distribution with 1 degrees of freedom . ab 1 . - - - a ...
Page 334
... degrees of freedom . Each of the random 72 variables √n ( ά – α ) / σ and √ ( c , − e ) 2 ( ß – B ) / o has a normal dis- tribution with zero mean and unit variance ; thus each of Q1 / 02 and Q2 / 02 has a chi - square distribution ...
... degrees of freedom . Each of the random 72 variables √n ( ά – α ) / σ and √ ( c , − e ) 2 ( ß – B ) / o has a normal dis- tribution with zero mean and unit variance ; thus each of Q1 / 02 and Q2 / 02 has a chi - square distribution ...
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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