## Question 4.2

Sigma<-matrix(c(2,1/sqrt(2),1/sqrt(2),1),2,2)
center<-c(0,2)
eigenSnvecs<-eigen(Sigma)$vectors
slope1<-eigenSnvecs[2,1]/eigenSnvecs[1,1]
intercept1<-center[2]-center[1]*slope1
slope2<-eigenSnvecs[2,2]/eigenSnvecs[1,2]
intercept2<-center[2]-center[1]*slope2
x1<-seq(-2,2,length=100)
x2<-seq(0,4,length=100)
plot(x1,x2,xlab=expression(x[1]),ylab=expression(x[2]),type="n",asp=1)
points(center[1],center[2],col=2,pch=19)
## Two main axes
abline(a=intercept1,b=slope1)
abline(a=intercept2,b=slope2)
npoints<-1000
r<-sqrt(qchisq(0.5,2))
theta<-seq(0, 2*pi, length = npoints)
v<-rbind(r*cos(theta), r*sin(theta))
## transform for points on ellipse
z<-backsolve(chol(solve(Sigma)),v)+center
## plot points
lines(t(z))

## Question 4.23

# part (a)
annualreturn<-c(-0.6, 3.1, 25.3, -16.8, -7.1, -6.2, 16.1, 25.2, 22.6, 26.0)
qqnorm(annualreturn)
qqline(annualreturn)

# part (b)

orderx<-sort(annualreturn)
n<-length(orderx)
px<-((1:n)-0.5)/n
qx<-qnorm(px)
rq<-cor(orderx,qx)

## Question 4.26

# part (a)

n<-10
x1<-c(1,2,3,3,4,5,6,8,9,11)
x2<-c(18.95, 19.00, 17.95, 15.54, 14.00, 12.95, 8.94, 7.49, 6.00, 3.99)
X<-cbind(x1,x2)
Xbar<-colMeans(X)
S<-cov(X)
Sinv<-solve(S)
djs<-diag(t(t(X)-Xbar)%*%Sinv%*%(t(X)-Xbar))

# part (b)

Sigma<-S
center<-Xbar
x10<-seq(0,12,length=100)
x20<-seq(3,20,length=100)
plot(x10,x20,xlab=expression(x[1]),ylab=expression(x[2]),type="n")
points(center[1],center[2],col=2,pch=19)
npoints<-1000
r<-sqrt(qchisq(0.5,2))
theta<-seq(0, 2*pi, length = npoints)
v<-rbind(r*cos(theta), r*sin(theta))
## transform for points on ellipse
z<-backsolve(chol(solve(Sigma)),v)+center
## plot points
lines(t(z))
points(x1,x2,cex=0.5)

# part (c)

qqplot(qchisq(ppoints(500), df = 2), djs,
 main = expression("Chi-square plot for" ~~ {chi^2}[2]))
qqline(djs,distribution=function(p) qchisq(p, df = 2))