Introduction to Mathematical Statistics |
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Page 25
... X to be a random variable of the discrete or continuous type . and ( a ) 0 ≤ F ( x ) ≤ 1 because 0 ≤ 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 to be a random variable of the discrete or continuous type . and ( a ) 0 ≤ F ( x ) ≤ 1 because 0 ≤ 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 ≤ ...
Page 90
... Pr ( X = 5 ) = 35e - 3 5 ! and , by Table I of the Appendix , Pr ( X = 5 ) = Pr ( X ≤ 5 ) – Pr ( X ≤ 4 ) - Pr ( X ≤ 4 ) = 0.101 , approximately . EXERCISES 3.17 . If the random variable X has a Poisson distribution such that Pr ( X ...
... Pr ( X = 5 ) = 35e - 3 5 ! and , by Table I of the Appendix , Pr ( X = 5 ) = Pr ( X ≤ 5 ) – Pr ( X ≤ 4 ) - Pr ( X ≤ 4 ) = 0.101 , approximately . EXERCISES 3.17 . If the random variable X has a Poisson distribution such that Pr ( X ...
Page 94
... ( x ) of this form is called a chi - square p.d.f. The mean and the variance of a chi - square distribution are μ αβ ... Pr ( c1 ≤ X ≤ © 2 ) = Pr ( X ≤ C2 ) - Pr ( X ≤ c1 ) , C1 ) 0. To compute such a probability , we need the since Pr ...
... ( x ) of this form is called a chi - square p.d.f. The mean and the variance of a chi - square distribution are μ αβ ... Pr ( c1 ≤ X ≤ © 2 ) = Pr ( X ≤ C2 ) - Pr ( X ≤ c1 ) , C1 ) 0. To compute such a probability , we need the since Pr ...
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A₁ A₂ Accordingly c₁ chi-square distribution complete sufficient statistic compute conditional p.d.f. confidence interval 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 hypothesis H₁ independent random variables integral joint p.d.f. k₁ Let the random Let X1 Let Y₁ likelihood ratio limiting distribution marginal p.d.f. moment-generating function mutually stochastically independent noncentral normal distribution order statistics p.d.f. of Y₁ P(A₁ Poisson distribution positive integer probability density functions probability set function 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 t₂ theorem unbiased statistic variance o² W₁ X₁ and X2 X₂ Y₂ Z₁ zero elsewhere μ₁ μ₂ Σ Σ σ²