lab3.r

# lab3.R			script file for lab3 calculations
#
# author: Eric Zivot
# created: September 17, 2008
# revised: July 6, 2012
#

# first install the mvtnorm package
options(digits=4)
library(mvtnorm)

#
# bivariate normal distribution
#

mu.x = 0.05
sig.x = 0.10
mu.y = 0.025
sig.y = 0.05

# simulate from bivariate normal with rho = 0.9
rho.xy = 0.9
sig.xy = rho.xy*sig.x*sig.y
Sigma.xy = matrix(c(sig.x^2, sig.xy, sig.xy, sig.y^2), 2, 2, byrow=TRUE)

# use the rmvnorm() function to simulate from bivariate normal
?rmvnorm
n = 100
set.seed(123)
xy.vals = rmvnorm(n, mean=c(mu.x, mu.y), sigma=Sigma.xy) 
head(xy.vals)

# scatterplot
plot(xy.vals[,1], xy.vals[,2], pch=16, cex=2, col="blue", 
     xlab="x", ylab="y")
title("Bivariate normal: rho=0.9")
abline(h=mu.y, v=mu.x)
segments(x0=0, y0=min(xy.vals[,2]), x1=0, y1=0, col="red")
segments(x0=min(xy.vals[,1]), y0=0, x1=0, y1=0, col="red")

# compute area under bivariate standard normal distribution
# Finc P( -00 < X < 0 and -00 < Y < 0)
?pmvnorm
pmvnorm(lower=c(-Inf, -Inf), upper=c(0, 0), mean=c(mu.x, mu.y), sigma=Sigma.xy)

# part (b)
# simulate from bivariate normal with rho=-0.9 
rhob.xy = -0.9
sigb.xy = rhob.xy * sig.x * sig.y
Sigmab.xy = matrix(c(sig.x^2, sigb.xy, sigb.xy, sig.y^2), 2, 2, byrow=TRUE)
xy.vals = rmvnorm(n, mean=c(mu.x, mu.y), sigma=Sigmab.xy)

# distribution scatterplot
plot(xy.vals[,1], xy.vals[,2], pch=16, cex=2, col="red", xlab="x", ylab="y")
title("Bivariate nurmal: rho=-0.9")
abline(h=mu.y, v=mu.x)

# compute probability of P(X < 0, Y < 0)
pmvnorm(lower=c(-Inf, -Inf), upper=c(0, 0), mean=c(mu.x, mu.y), sigma=Sigmab.xy)


# part (c)
# simulate from bivariate normal with rho=-0.9 
rhoc.xy = 0
sigc.xy = rhoc.xy * sig.x * sig.y
Sigmac.xy = matrix(c(sig.x^2, sigc.xy, sigc.xy, sig.y^2), 2, 2, byrow=TRUE)
xy.vals = rmvnorm(n, mean=c(mu.x, mu.y), sigma=Sigmac.xy)

# distribution scatterplot
plot(xy.vals[,1], xy.vals[,2], pch=16, cex=2, col="green", xlab="x", ylab="y")
title("Bivariate nurmal: rho=0")
abline(h=mu.y, v=mu.x)

# compute probability of P(X < 0, Y < 0)
pmvnorm(lower=c(-Inf, -Inf), upper=c(0, 0), mean=c(mu.x, mu.y), sigma=Sigmac.xy)