Introduction to Mathematical Statistics |
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Page 20
... Pr ( XE A ) can be written as = rb Pr ( a < X < b ) = so ƒ ( x ) dx . { x ; x = a } , then Moreover , if A = a P ( A ) = Pr ( X = A ) = Pr ( X = a ) = √a f ( x ) dx = = 0 , since the integral ( a f ( x ) dx is defined in calculus to be ...
... Pr ( XE A ) can be written as = rb Pr ( a < X < b ) = so ƒ ( x ) dx . { x ; x = a } , then Moreover , if A = a P ( A ) = Pr ( X = A ) = Pr ( X = a ) = √a f ( x ) dx = = 0 , since the integral ( a f ( x ) dx is defined in calculus to be ...
Page 25
... Pr ( X ≤ x ) ≤ 1 . ( b ) F ( x ) is a nondecreasing function of x . For , if x ' < x " , then That is , { x ; x ≤ x " } = { x ; x ≤ x ' } \ { x ; x ' < x ≤ x " } , Pr ( X ≤ x " ) = Pr ( X ≤ x ' ) + Pr ( x ' < X ≤ x " ) . F ( x ...
... Pr ( X ≤ x ) ≤ 1 . ( b ) F ( x ) is a nondecreasing function of x . For , if x ' < x " , then That is , { x ; x ≤ x " } = { x ; x ≤ x ' } \ { x ; x ' < x ≤ x " } , Pr ( X ≤ x " ) = Pr ( X ≤ x ' ) + Pr ( x ' < X ≤ x " ) . F ( x ...
Page 71
... Pr ( a < X1 < b ) Pr ( c < X2 < d ) 1 for every a < b and c < d , where a , b , c , and d are constants . 1 Proof . From the stochastic independence of X1 and X2 , the joint p.d.f. of X , and X2 is f1 ( x1 ) ƒ2 ( x2 ) . Accordingly , in ...
... Pr ( a < X1 < b ) Pr ( c < X2 < d ) 1 for every a < b and c < d , where a , b , c , and d are constants . 1 Proof . From the stochastic independence of X1 and X2 , the joint p.d.f. of X , and X2 is f1 ( x1 ) ƒ2 ( x2 ) . Accordingly , in ...
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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