Transformations and random variate generation: generalised ratio-of-uniform methods

The basic ingredient of random variate generation is, of course, the uniform random number and, in particular, the ability to generate points uniformly over some twodimensional region. One can, for instance, choose such a region or regions to be easy or efficient to generate from, and then adapt the result to the region of interest which is that beneath the density curve, as done most famously, for example, in rejection methods. An alternative approach, explored here, is to transform from the easier uniform region to the desired variate. Not only is the inversion method an obvious example of this, but the approach also covers the more mysterious ratio-of-uniforms method and generalisations thereof which we are able to generalise yet further. Our aim is largely explanatoryjpedagogical rather than the explicit provision of new methods