### Example of a joint density.

x <- seq(0,5,0.2)

y <- x

f <- function(x,y){ 2*exp(-x)*exp(-2*y) }

z <- outer(x,y,f)

par(mfrow=c(2,2))

persp(z)

persp(z,eye=c(5,-1,3))

persp(z,eye=c(-1,5,3))

persp(z,eye=c(5,5,3))

### Example of plotting the joint normal distribution.

x <- seq(-2,2,0.1)

y <- seq(-2,2,0.1)

par(mfrow=c(2,2))

bvn <- function(x,y){

mu1 <- 0

sigma1 <- 1

mu2 <- 0

sigma2 <- 1

rho <- 0

((2*pi*sigma1*sigma2*sqrt(1-rho^2))^(-1))*

exp( -(2*((1-rho^2))^(-1)) * ( (x-mu1)^2/sigma1^2 + (y-mu2)^2/sigma2^2

- ( (2*rho*(x-mu1)*(y-mu2) )/ (sigma1*sigma2) ) ) )

}

z1 <- outer(x,y,bvn)

persp(x,y,z1)

bvn <- function(x,y){

mu1 <- 0

sigma1 <- 1

mu2 <- 0

sigma2 <- 1

rho <- 0.3

((2*pi*sigma1*sigma2*sqrt(1-rho^2))^(-1))*

exp( -(2*((1-rho^2))^(-1)) * ( (x-mu1)^2/sigma1^2 + (y-mu2)^2/sigma2^2

- ( (2*rho*(x-mu1)*(y-mu2) )/ (sigma1*sigma2) ) ) )

}

z2 <- outer(x,y,bvn)

persp(x,y,z2)

bvn <- function(x,y){

mu1 <- 0

sigma1 <- 1

mu2 <- 0

sigma2 <- 1

rho <- 0.6

((2*pi*sigma1*sigma2*sqrt(1-rho^2))^(-1))*

exp( -(2*((1-rho^2))^(-1)) * ( (x-mu1)^2/sigma1^2 + (y-mu2)^2/sigma2^2

- ( (2*rho*(x-mu1)*(y-mu2) )/ (sigma1*sigma2) ) ) )

}

z3 <- outer(x,y,bvn)

persp(x,y,z3)

bvn <- function(x,y){

mu1 <- 0

sigma1 <- 1

mu2 <- 0

sigma2 <- 1

rho <- -0.9

((2*pi*sigma1*sigma2*sqrt(1-rho^2))^(-1))*

exp( -(2*(1-rho^2)^(-1)) * ( (x-mu1)^2/sigma1^2 + (y-mu2)^2/sigma2^2

- ( (2*rho*(x-mu1)*(y-mu2) )/ (sigma1*sigma2) ) ) )

}

z4 <- outer(x,y,bvn)

persp(x,y,z4)

par(mfrow=c(2,2))

contour(x,y,z1)

contour(x,y,z2)

contour(x,y,z3)

contour(x,y,z4)