Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig
3.5. The Multivariate Normal Distribution 205 This shows the equivalence of the bivariate normal pdf notation, (3.5.1), and the ...
206 Some Special Distributions 3.5.3 ∗Applications........................... In this section, we consider several applications ...
3.5. The Multivariate Normal Distribution 207 What about the other components,Y 2 ,...,Yn? As the following theorem shows, they ...
208 Some Special Distributions ellipmake(p=.95,b=matrix(c(1,.75,.75,1),nrow=2),mu=c(5,2)). This R function can be found at the s ...
3.5. The Multivariate Normal Distribution 209 3.5.12.LetXandY have a bivariate normal distribution with parametersμ 1 = μ 2 =0,σ ...
210 Some Special Distributions (a)Find the total variation ofX. (b)Find the principal component vectorY. (c)Show that the first ...
3.6.t-andF-Distributions 211 Define a new random variableTby writing T= W √ V/r . (3.6.1) The transformation technique is used t ...
212 Some Special Distributions aspt(2.0,15), while the commandqt(.975,15)returns the 97.5th percentile of this distribution. The ...
3.6.t-andF-Distributions 213 We define the new random variable W= U/r 1 V/r 2 and we propose finding the pdfg 1 (w)ofW. The equa ...
214 Some Special Distributions In terms of R computation, the commandpf(2.50,3,8)computes to the value 0 .8665 which is the prob ...
3.6.t-andF-Distributions 215 (d)The random variable T= X−μ S/ √ n (3.6.9) has a Studentt-distribution withn− 1 degrees of freedo ...
216 Some Special Distributions By part (b), the two terms on the right side of the last equation are independent. Further, the s ...
3.6.t-andF-Distributions 217 (e)Xhas at-distribution with 30 degrees of freedom. 3.6.6.In expression (3.4.13), the normal locati ...
218 Some Special Distributions 3.6.14.Show that Y= 1 1+(r 1 /r 2 )W , whereWhas anF-distribution with parametersr 1 andr 2 , has ...
3.7.∗Mixture Distributions 219 a weighted average ofμ 1 ,μ 2 ,...,μk, and the variance equals var(X)= ∑k i=1 pi ∫∞ −∞ (x−μ)^2 fi ...
220 Some Special Distributions The mixture of distributions is sometimes calledcompounding.Moreover,it does not need to be restr ...
3.7.∗Mixture Distributions 221 for−∞<x<∞, 0 <θ<∞. Therefore, the marginal (unconditional) pdfh(x) ofXis found by int ...
222 Some Special Distributions Example 3.7.4.In this example, we develop by compounding a heavy-tailed skewed distribution. Assu ...
3.7.∗Mixture Distributions 223 EXERCISES 3.7.1.SupposeY has a Γ(α, β) distribution. LetX=eY. Show that the pdf of Xis given by e ...
224 Some Special Distributions 3.7.10.For the Burr distribution, show that E(Xk)= 1 βk/τ Γ ( α− k τ ) Γ ( k τ +1 )/ Γ(α), provid ...
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