Transformation of Uniformly Distributed Random Numbers

• Based on standard pseudo-random numbers $u_1,u_2\ldots$ that can be generated by methods like the linear congruential generator
  • it is possible to generate pseudo-random numbers $x_1,x_2\ldots$ that can be regarded as realizations of random variables $X_1,X_2\ldots$ having other than uniform distributions.
  • Examples are realizations $x_1,x_2,\ldots$ of exponentially, Poisson, binomially or normally distributed random variables $X_1,X_2,\ldots$.
• For this purpose one can apply algorithms like the so-called inversion method and rejection-sampling, whose basic ideas will be explained by some examples.
• A much more comprehensive discussion of these algorithms can be found e.g. in
  • L. Devroye (1986) Nonuniform Random Variate Generation. Springer, New York,
  • G.S. Fishman (1996) Monte Carlo: Concepts, Algorithms and Applications. Springer, New York,
  • C.P. Robert and G. Casella (1999) Monte Carlo Statistical Methods. Springer, New York.