Introduction to Mathematical StatisticsThe fifth edition of text offers a careful presentation of the probability needed for mathematical statistics and the mathematics of statistical inference. Offering a background for those who wish to go on to study statistical applications or more advanced theory, this text presents a thorough treatment of the mathematics of statistics. |
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Page 44
Robert V. Hogg, Allen Thornton Craig. ( b ) f ( x ) = 12x2 ( 1 − x ) , 0 < x < 1 , zero elsewhere . ( c ) f ( x ) = ( { } ) x2e ̄ * , 0 < x < ∞ , zero ... x 44 Probability and Distributions [ Ch . 1 Properties of the Distribution Function.
Robert V. Hogg, Allen Thornton Craig. ( b ) f ( x ) = 12x2 ( 1 − x ) , 0 < x < 1 , zero elsewhere . ( c ) f ( x ) = ( { } ) x2e ̄ * , 0 < x < ∞ , zero ... x 44 Probability and Distributions [ Ch . 1 Properties of the Distribution Function.
Page 47
... F ( x ) is a nondecreasing function of x , which is everywhere continuous from the right and has F ( − ∞ ) = 0 , F ( ∞ ) = 1. The probability Pr ( a < X ≤ b ) is equal to the difference F ( b ) — F ( a ) . If ... Distribution Function 47.
... F ( x ) is a nondecreasing function of x , which is everywhere continuous from the right and has F ( − ∞ ) = 0 , F ( ∞ ) = 1. The probability Pr ( a < X ≤ b ) is equal to the difference F ( b ) — F ( a ) . If ... Distribution Function 47.
Page 237
... x ≤ 2 , ∞18 = 1 , x > 2 . F ( x ) = 0 , x < 2 , = x ≥ 2 , is a distribution function , and since lim F ( x ) = F ( x ) at all points of continuity of F ( x ) , the sequence X1 , X2 , X3 , ... converges in distribution to a random ...
... x ≤ 2 , ∞18 = 1 , x > 2 . F ( x ) = 0 , x < 2 , = x ≥ 2 , is a distribution function , and since lim F ( x ) = F ( x ) at all points of continuity of F ( x ) , the sequence X1 , X2 , X3 , ... converges in distribution to a random ...
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A₁ A₂ Accordingly approximate best critical region C₁ C₂ chi-square distribution complete sufficient statistic conditional p.d.f. confidence interval Consider continuous type converges in probability correlation coefficient critical region defined degrees of freedom denote a random discrete type distribution function F(x distribution with mean distribution with p.d.f. distribution with parameters dx₁ equation Example Exercise Find the p.d.f. g₁(y₁ gamma distribution given Hint hypothesis H₁ independent random variables integral Let the random Let X1 Let Y₁ limiting distribution marginal p.d.f. matrix moment-generating function order statistics p.d.f. of Y₁ P(C₁ p₁ Poisson distribution positive integer probability density functions probability set function R₁ r₂ random experiment random sample respectively sample space Section Show significance level simple hypothesis subset sufficient statistic t-distribution t₂ theorem unbiased estimator variance o² W₁ X₁ X₂ Y₁ Y₂ zero elsewhere μ₁ Σ Σ σ²