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IID Statistics and Random Sampling. In statistics, we commonly deal with random samples. A random sample can be thought of as a set of objects that are chosen randomly. Or, more formally, it's "a sequence of independent, identically distributed (IID) random variables".The standard version of the central limit theorem, first proved by the French mathematician Pierre-Simon Laplace in 1810, states that the sum or average of an infinite sequence of independent and identically distributed random variables, when suitably rescaled, tends to a normal distribution. Laplace transform of a sum Let X and Y be independent random variables with L-transforms f∗ X(s) and f∗ Y(s). f∗ X+Y(s) = E[e. −s(X+Y)] = E[e−sXe−sY] = E[e−sX]E[e−sY] (independence) = f∗ X(s)f∗ Y(s) f∗ X+Y(s) = f. ∗ X(s)f∗ Y(s) J. Virtamo 38.3143 Queueing Theory / Continuous Distributions 3. May 19, 2009 · Abstract: Accurate closed-form approximations to the sum of independent identically distributed eta-mu and kappa-mu random variables are provided. The proposed approximations turn out to be simple, precise, and useful for obtaining important performance metrics of communications systems where sums of variates arise.