I have a distribution of discrete random variables Q and the random
variables q drawn from Q are positive integers. The distribution Q is
fully known, so I know the probability Pr{q=1}, Pr{q=2}, and so on.

I am trying to compute the distribution of the product of k variables
PQ=q_1 * q_2 * q_3 ... q_k, where q_i are all independent variables,
drawn from Q (i.e., i.i.d., drawn from Q)

How do I compute the distribution of the random variable PQ?  In other
words, how do I compute the probability  Pr{PQ=1}, Pr{PQ=2} and so on?

Please do not assume that k is large; I know that for large k the PQ
has a log-normal distribution. I am interested in small values of k.

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Clarification of Question by ipeirotis-ga on 05 Nov 2005 15:49 PST

Just to clarify: 

1. I am not interested in "brute-force" approaches. I am looking for
something computationally efficient.

2. I am aware of the necessary transformations (Mellin transformation)
for the case of continuous variables.