Introduction to Mathematical Statistics |
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Page 69
... ( x1 ) ƒ2 ( x2 ) means a function that is positive on a product space . That is , if f ( x ) and ƒ2 ( x2 ) are positive on , and only on , the respective spaces 1 and 2 , then the product of f1 ( x1 ) and f2 ( x2 ) is positive on , and ...
... ( x1 ) ƒ2 ( x2 ) means a function that is positive on a product space . That is , if f ( x ) and ƒ2 ( x2 ) are positive on , and only on , the respective spaces 1 and 2 , then the product of f1 ( x1 ) and f2 ( x2 ) is positive on , and ...
Page 70
... ( x1 , x2 ) = f1 ( x1 ) ƒ2 ( x2 ) , where ƒ1 ( x1 ) and ƒ2 ( x2 ) are the marginal probability density functions of X1 and X2 respectively . Thus , the condition f ( x1 , X2 ) = g ( x1 ) h ( x2 ) is fulfilled . 1 Conversely , if f ( x1 ...
... ( x1 , x2 ) = f1 ( x1 ) ƒ2 ( x2 ) , where ƒ1 ( x1 ) and ƒ2 ( x2 ) are the marginal probability density functions of X1 and X2 respectively . Thus , the condition f ( x1 , X2 ) = g ( x1 ) h ( x2 ) is fulfilled . 1 Conversely , if f ( x1 ...
Page 72
... X1 and X2 have the marginal probability density functions ƒ1 ( x1 ) and ƒ2 ( x2 ) respectively . The expected value of the product of a function u ( X1 ) of X1 alone and a function v ( X2 ) of X2 alone is , subject to their existence ...
... X1 and X2 have the marginal probability density functions ƒ1 ( x1 ) and ƒ2 ( x2 ) respectively . The expected value of the product of a function u ( X1 ) of X1 alone and a function v ( X2 ) of X2 alone is , subject to their existence ...
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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 F distribution function F(x given hypothesis H₁ 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 Poisson distribution positive integer probability density functions quadratic form random experiment random interval random sample random variables X1 respectively Show significance level simple hypothesis statistic Y₁ stochastically independent random sufficient statistic theorem unbiased statistic variables X₁ variance o² W₁ X₁ X₁ and X2 X₂ x²(n Y₂ Z₁ zero elsewhere μ₁ σ²