Question

sketch a contour plot for the joint pdf of X and Y where X~ = normal (3,1)  independent of Y~ Normal (0,4)

Answer 1

The contour plot of the joint pdf of and is as follows, where and independently:

The following R code was used for the plot:

x = seq(-5,5,0.01)
y = seq(-5,5,0.01)
n = length(x)
z = numeric(n*n)
dim(z) = c(n,n)
for(i in 1:n)
{
  for(j in 1:n)
  {
    z[i,j] = dnorm(x[i],mean = 3, sd = 1)*dnorm(y[j], mean = 0, sd = 2)
  }
}

contour(x,y,z, xlim = c(-1,6))
points(x = 3, y = 0, pch = 16)
abline(v = 3)
abline(0,0)