I need to implement a program that generate a sample from a 3 variated $pdf$ using the Metropolis algorithm (and its variations). I was thinking to use a 3-variated normal distribution as my proposal distribution, but I am having some difficult and doubts doing it. For example, I've seen this simulation in R: http://blog.abhranil.net/2014/02/08/r-code-for-multivariate-random-walk-metropolis-hastings-sampling/
1) He uses metropolis random walk with mean $X^t$ and a given covariance matrix. It seens he uses the fact that this distribution is symmetric. I know it is a normal, but is it always symmetric?
2) What would be better: Draw from a trivariated normal directly or draw each parameter from univariated normal? In the last case, we can't adjust covariance, so it looks less efficient and I don't even know it is valid.
Thanks in advance!