Introduction to Mathematical Statistics |
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Page 112
... one or more random variables is called the change of variable technique ... transformation from a to y , x and we say that the transformation maps the space onto the space B = { y ; y = 0 , 4 , 8 , 12 , ... } . The space is obtained by ...
... one or more random variables is called the change of variable technique ... transformation from a to y , x and we say that the transformation maps the space onto the space B = { y ; y = 0 , 4 , 8 , 12 , ... } . The space is obtained by ...
Page 113
... each of which f ( x ) > 0 , and let y u ( x ) define a one - to - one transformation that maps A onto B. If we solve y u ( x ) for x in terms of y , say , x = w ( y ) , then for each y B , we have x = w ( y ) A. Consider the random ...
... each of which f ( x ) > 0 , and let y u ( x ) define a one - to - one transformation that maps A onto B. If we solve y u ( x ) for x in terms of y , say , x = w ( y ) , then for each y B , we have x = w ( y ) A. Consider the random ...
Page 118
... < 1 } , which the one - to - one transformation y -2 ln x maps onto B = { y ; 0 < y < ∞o } . The Jacobian of the transformation is J = dx dy = w ' ( y ) = V / 2 . Accordingly , the p.d.f. g ( y ) of Y - 2 In X is g ( y ) = f ( e - v / 2 ) ...
... < 1 } , which the one - to - one transformation y -2 ln x maps onto B = { y ; 0 < y < ∞o } . The Jacobian of the transformation is J = dx dy = w ' ( y ) = V / 2 . Accordingly , the p.d.f. g ( y ) of Y - 2 In X is g ( y ) = f ( e - v / 2 ) ...
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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