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 33
... x x = = 1 , 2 , 3 , . . . , elsewhere . From such a function , we see that the space is clearly the set of positive integers which is a discrete set of points . Thus the corresponding random ... Random Variables of the Discrete Type 33.
... x x = = 1 , 2 , 3 , . . . , elsewhere . From such a function , we see that the space is clearly the set of positive integers which is a discrete set of points . Thus the corresponding random ... Random Variables of the Discrete Type 33.
Page 164
... x x = 0 , 1 , 2 , 3 , ye B , = 0 elsewhere . = We seek the p.d.f. g ( y ) of the random variable Y = X2 . The transformation y = u ( x ) x2 maps = { x : x = 0 ... variables Y1 = u1 164 Distributions of Functions of Random Variables [ Ch . 4.
... x x = 0 , 1 , 2 , 3 , ye B , = 0 elsewhere . = We seek the p.d.f. g ( y ) of the random variable Y = X2 . The transformation y = u ( x ) x2 maps = { x : x = 0 ... variables Y1 = u1 164 Distributions of Functions of Random Variables [ Ch . 4.
Page 447
... X 2 + ... + X32 ) - 2 ( x , x 2 + ··· + X , Xx + ··· + Xn - 1 Xn ) ... n is a quadratic form in the n variables X1 , X2 , . . . , X. If the sample arises from a distribution that is N ( μ , o2 ) , we know that the random variable nS2 ...
... X 2 + ... + X32 ) - 2 ( x , x 2 + ··· + X , Xx + ··· + Xn - 1 Xn ) ... n is a quadratic form in the n variables X1 , X2 , . . . , X. If the sample arises from a distribution that is N ( μ , o2 ) , we know that the random variable nS2 ...
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Common terms and phrases
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