Question

Suppose I have a bivariate discrete distribution, i.e. a table of probability values P(X=i,Y=j), for i=1,…n and j=1,…m. How do I generate a random sample (X_k,Y_k), k=1,…N from such distribution? Maybe there is a ready R function like:

sample(100,prob=biprob)

where biprob is 2 dimensional matrix?

One intuitive way to sample is the following. Suppose we have a data.frame

dt=data.frame(X=x,Y=y,P=pij)

Where x and y come from

expand.grid(x=1:n,y=1:m)

and pij are the P(X=i,Y=j).

Then we get our sample (Xs,Ys) of size N, the following way:

set.seed(1000) 
Xs <- sample(dt$X,size=N,prob=dt$P)
set.seed(1000)
Ys <- sample(dt$Y,size=N,prob=dt$P)

I use set.seed() to simulate the “bivariateness”. Intuitively I should get something similar to what I need. I am not sure that this is correct way though. Hence the question 🙂

Another way is to use Gibbs sampling, marginal distributions are easy to compute.

I tried googling, but nothing really relevant came up.

Answer 1

You are almost there. Assuming you have the data frame dt with the x, y, and pij values, just sample the rows!

dt <- expand.grid(X=1:3, Y=1:2)
dt$p <- runif(6)
dt$p <- dt$p / sum(dt$p)  # get fake probabilities
idx <- sample(1:nrow(dt), size=8, replace=TRUE, prob=dt$p)
sampled.x <- dt$X[idx]
sampled.y <- dt$Y[idx]