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 209
Robert V. Hogg, Allen Thornton Craig. then the moment - generating function of Y = 1⁄2 a ; X11 i = 1 where a1 , a2 , · ak are real constants , is 9 n My ( t ) = [ ] M1 ( a , t ) . i = 1 Proof . The m.g.f. of Y is given by My ( t ) = E ...
Robert V. Hogg, Allen Thornton Craig. then the moment - generating function of Y = 1⁄2 a ; X11 i = 1 where a1 , a2 , · ak are real constants , is 9 n My ( t ) = [ ] M1 ( a , t ) . i = 1 Proof . The m.g.f. of Y is given by My ( t ) = E ...
Page 213
... 2abpo ̧σ2 + b2o2 ) , where a , b , and c are constants . Hint : Use the m.g.f. M ( t ,, t2 ) of X and Y to find the m.g.f. of Z. 4.88 . Let X and Y have a bivariate normal Sec . 4.7 The Moment - Generating - Function Technique 213.
... 2abpo ̧σ2 + b2o2 ) , where a , b , and c are constants . Hint : Use the m.g.f. M ( t ,, t2 ) of X and Y to find the m.g.f. of Z. 4.88 . Let X and Y have a bivariate normal Sec . 4.7 The Moment - Generating - Function Technique 213.
Page 243
Robert V. Hogg, Allen Thornton Craig. 5.3 Limiting Moment - Generating Functions To find the limiting distribution function of a random variable Y , by use of the definition of limiting distribution function obviously requires that we ...
Robert V. Hogg, Allen Thornton Craig. 5.3 Limiting Moment - Generating Functions To find the limiting distribution function of a random variable Y , by use of the definition of limiting distribution function obviously requires that we ...
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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. gamma distribution given H₁ Hint hypothesis H independent random variables integral Let the random Let X1 Let Y₁ limiting distribution marginal p.d.f. matrix mean µ moment-generating function order statistics p.d.f. of Y₁ P(C₁ p₁ percent confidence interval Poisson distribution positive integer probability density functions probability set function R₁ random experiment random sample respectively sample space Section Show significance level simple hypothesis subset sufficient statistic t-distribution t₂ theorem unbiased estimator variance o² X₁ X₂ Y₁ Y₂ zero elsewhere μ₁ σ²