Introduction to Mathematical Statistics |
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Page 58
... conditional mean " and the " conditional variance " of X2 , given X1 - = 1. Of course we have E { [ X2 – E ( X2 | x1 ) ] 2 | x1 } = E ( X2 | x1 ) − [ E ( X2 | x1 ) ] 2 - from an earlier result . In like manner , the conditional ...
... conditional mean " and the " conditional variance " of X2 , given X1 - = 1. Of course we have E { [ X2 – E ( X2 | x1 ) ] 2 | x1 } = E ( X2 | x1 ) − [ E ( X2 | x1 ) ] 2 - from an earlier result . In like manner , the conditional ...
Page 60
... conditional p.d.f. If f1 ( x1 ) > 0 , the symbol f ( x2 , . . . , xnx1 ) is defined by the relation f ( x2 , ... , xn x1 ) = f ( x1 , x2 , ... , Xn ) . f1 ( x1 ) = n = and f ( x2 , ... , xnx1 ) is called the joint conditional p.d.f. of ...
... conditional p.d.f. If f1 ( x1 ) > 0 , the symbol f ( x2 , . . . , xnx1 ) is defined by the relation f ( x2 , ... , xn x1 ) = f ( x1 , x2 , ... , Xn ) . f1 ( x1 ) = n = and f ( x2 , ... , xnx1 ) is called the joint conditional p.d.f. of ...
Page 104
... conditional p.d.f. of Y , given X = x , is itself normal with mean μ2 + p ( σ2 / σ1 ) ( x — μ1 ) and variance p2 ) . Thus , with a bivariate normal distribution , the conditional mean of Y , given X = x , is linear in x and is given by ...
... conditional p.d.f. of Y , given X = x , is itself normal with mean μ2 + p ( σ2 / σ1 ) ( x — μ1 ) and variance p2 ) . Thus , with a bivariate normal distribution , the conditional mean of Y , given X = x , is linear in x and is given by ...
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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