R Studio, Multivariate Normal Distribution, Ellipse, Confidence Interval, Cholesky decomposition

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STAT821 Assignment 1 Solution with R Codes.

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Answer 1a) The Cholesky decomposition approach takes an original X variable and uses theCholesky transformation to create a new, correlated Y variable.b) The R program is given below.S<-matrix(c(1,1,1,1,3,2,1,2,2),nrow=3)R<-cov2cor(S)U<-t(chol(R))c) The 100 random numbers are obtained below.S<-matrix(c(1,1,1,1,3,2,1,2,2),nrow=3)R<-cov2cor(S)U<-t(chol(R))nvars<-dim(U)[1]numobs<-100set.seed(1)random.normal <- matrix(rnorm(nvars*numobs,0,1), nrow=nvars, ncol=numobs)X<-U%*%random.normalnewX<-t(X)The box plot for each variable is given below.From the above plot it can be observed that the first box plot has the maximumvariation and also has one outlier. For the second box plot, the variation of theobservations has decreased but still has one outlier. In the final boxplot, the variationis the least and there are no outliers, that is, the third box plot is more approximate tothe normal distribution.Answer 2a) If X~N3 (, ), then any linear combination of X, that is, a X ~ N1 a , aa . HerePage | 1211 1 1 a 3 , 2 and 1 3 2 . Then 2 1 2 2 0 11 1 1 2 a 2 3 0 2 8, and aa 2 3 0 1 3 2 3 19.1 2 2 0 2 Therefore 2x1 3x2 ~ N1(8, 19).> a<-matrix(
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