When would it be computationally inefficient to sample from a distribution? I am reading some stuff about MCMC simulation and using that as a method to sample from a distribution.  I understand that MCMC algos can be used to approximate a distribution when we are unable to obtain a closed form solution for the PDF.  However, the book also mentions that we can use MCMC simulation when it is not "computationally efficient" to directly sample from the distribution.
What would cause it to be computationally inefficient to sample directly from a distribution?  What would be a good example?
 A: (1) Simple Gibbs example: Sample from bivariate normal distribution
with $\mu_x = \mu_y= 0, \sigma_x = \sigma_y = 1, \rho = .8.$
This can be done as follows using a Gibbs sampler, based on
$X|Y=y \sim \mathsf{Norm}(\rho y, \sqrt{1-\rho^2})$ and
$Y|X=x \sim \mathsf{Norm}(\rho x, \sqrt{1-\rho^2}).$ 
One cycles back and forth between the two conditional distributions to
get a Markov chain with the desired bivariate distribution as its limiting distribution. The burn-in period used below is at half of the
iterations.
set.seed(1029); m=100000
xc = yc = numeric(m)
rho=.8; sgm=sqrt(1-rho^2)
xc[1]=-3; yc[1]=3  # start step
for(i in 2:m) {
  xc[i] = rnorm(1,rho*yc[i-1],sgm)
  yc[i] = rnorm(1,rho*xc[i],sgm)  }
x = xc[(m/2+1):m]; y = yc[(m/2+1):m]
round(c(mean(x),mean(y), sd(x),sd(y), cor(x,y)),3)
[1] 0.001 0.004 1.004 1.003 0.800


(2) Chib and Greenberg (1994), Understanding the Metropolis-Hastings Algorithm, The American Statistician, 49 327-335, generate a correlated bivariate normal distribution as a simple illustration
of the M-H algorithm.
Any pedagogical purposes aside, both (1) and (2) are inefficient for this
simple simulation because @glen-b's method is simpler.
