Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig
3.3. TheΓ,χ^2 ,andβDistributions 185 3.3.20.Determine the constantcso thatf(x)=cx(3−x)^4 , 0 <x<3, zero elsewhere, is a pd ...
186 Some Special Distributions (b)Ifr(x)=cebx,wherecandbare positive constants, show thatX has a Gompertzcdf given by F(x)= { 1 ...
3.4. The Normal Distribution 187 This integral exists because the integrand is a positive continuous function that is bounded by ...
188 Some Special Distributions Upon evaluating these derivatives att= 0, the mean and variance ofZare E(Z)=0 and Var(Z)=1. (3.4. ...
3.4. The Normal Distribution 189 x f(x) 2 1 3 – 2 – + + 2 + 3 Figure 3.4.1:The normal densityf(x), (3.4.6). points of inflecti ...
190 Some Special Distributions its pdf is displayed in Figure 3.4.2. Common notation for the cdf ofZis P(Z≤z)=Φ(z)=dfn ∫z 0 1 √ ...
3.4. The Normal Distribution 191 normtab <- function(){ za <- seq(0.00,3.59,.01); pz <- t(matrix(round(pnorm(za),digits ...
192 Some Special Distributions whereeis a random variable (often called random error) with aN(0,σ^2 ) distribu- tion. Conversely ...
3.4. The Normal Distribution 193 Theorem 3.4.2.LetX 1 ,...,Xnbe independent random variables such that, for i=1,...,n,Xihas aN(μ ...
194 Some Special Distributions see Exercise 3.4.24. Upon differentiating (3.4.15), the pdf ofWis fW(w)=φ(w)(1− )+φ(w/σc) σc , (3 ...
3.4. The Normal Distribution 195 3.4.8.Evaluate ∫ 3 2 exp[−2(x−3) (^2) ]dx. 3.4.9.Determine the 90th percentile of the distribut ...
196 Some Special Distributions 3.4.21.Letf(x)andF(x) be the pdf and the cdf, respectively, of a distribution of the continuous t ...
3.4. The Normal Distribution 197 (a)Clearlyf(x;α)>0foallx. Show that the pdf integrates to 1 over (−∞,∞). Hint:Start with ∫∞ ...
198 Some Special Distributions 3.5 TheMultivariateNormalDistribution In this section we present the multivariate normal distribu ...
3.5. The Multivariate Normal Distribution 199 elliptical. IfXandY are independent then these contours are circular. The in- tere ...
200 Some Special Distributions Consider the random vectorZ=(Z 1 ,...,Zn)′,whereZ 1 ,...,Znare iidN(0,1) random variables. Then t ...
3.5. The Multivariate Normal Distribution 201 Note thatΣ^1 /^2 is symmetric and positive semi-definite. SupposeΣis positive defi ...
202 Some Special Distributions In Section 3.5.1, we discussed the contours of the bivariate normal distribution. We now extend t ...
3.5. The Multivariate Normal Distribution 203 whereX 2 is of dimensionp=n−m. In the same way, partition the mean and covariance ...
204 Some Special Distributions following proof shows, we can combine the results of Theorems 3.5.2 and 3.5.3 to obtain the follo ...
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