The joint pdf of two INDEPENDENT variables is the product of their individual pdf's...

For dependent variables it depends on the dependence?

For the multivariate Normal distribution the argument of the exponential is expressible in matrix notation as:

XT.inverse of the (Var-Covariance matrix).X,1/2

where "." means multiplication, X is the column vector containing all the variables and XT is the transpose of X. If all the variables are independent the the matrix is diagonal and the argument of the exponential is simply a sum over the squared deviations from the mean divided by the respective variances...or as Glenn points out simply the product of the individual pdf's. Otherwise the cross terms in the matrix adjust for the inter dependence among the variables.

Sandipan, This is hard to say w/o a chalkboard. Check out Wikipedia's discussion of the multivariate normal distribution for clarity.

Dear Francis I had asked about discrete Random Variables. Suppose I am having 3 discrete random variables' distribution like

X1 X2 X3

===========

1 1 1

1 1 2

... ... ...

===========

For any variable set it will be (n)^m, n=no. of levels and m= no. of variables.

When m is too big like 20 and we dont have the full sample set then how to proceed?

Full sample set means all possible combinations.

If all 3 are discrete and they are statistically dependent, there's no way I guess but to consider all possible combinations of possible values of X1, X2, X3. But you must have some kind of idea about the dependence if you can not assume that the outcomes are equally likely to happen.

I think this is the place to start reading. You are heading into some serious mathematics.

http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471128449.html