Problem: If the random variable Y is a monotonic increasing function of random variable X such that Y g X and X g Y (inversion of g . requires it to be monotonic) then, given the cdf F X x and the function g . , what are F Y y and f Y y , the cdf and pdf of Y ?

Solution: If g . is monotonic increasing, the cdf of Y is given by F Y y Y y g X g x X x F X x where y g x . The pdf of Y may be found as follows: f Y y y F Y y y F X x x F X x y x f X x y x Defining x y g x and f Y y f X x g x This relation is illustrated in , using a geometric construction to relate f Y to f X via Y g X . The area under each of the pdfs between a given pair of dashed lines must be the same, because the probability of being in a given range of X must be the same as the probability of being in the equivalent range of Y.

Illustration of monotonic mapping of pdfs by plotting f Y y rotated by 90°. The non-linearity in this case is g X 0.4 X 2 X 0.4 , which is monotonic for -1 X 1 . If g . is monotonic decreasing (instead of increasing), then becomes F Y y g X g x X x 1 F X x and by a similar argument we find that f Y y f X x g x In principle, any non-monotonic function g . can be split into a finite number of monotonic sections and in that case the pdf result can be generalized to f Y y i x x i f X x g x where the x i are all the solutions of g x y at any given y . However care is needed in this case, because if g x is smooth then g x will become zero at the section boundaries and so f Y y will tend to infinity at these points.

Example - Generation of a Gaussian pdf from a uniform pdf If X has a uniform pdf from 0 to 1 (and zero elsewhere), and we wish to generate Y using Y g X such that Y has a Gaussian (normal) pdf of unit variance and zero mean, what is the required function g . ? (This function is often needed in computers, because standard random number generators tend to have uniform pdfs, while simulation of noise from the real world requires Gaussian pdfs.) For these pdfs: f X x 1 0 x 1 0 f Y y 1 2 y 2 2 The corresponding cdfs are F X x u x f X u 0 x 0 x 0 x 1 1 x 1 F Y y u y f Y u From our previous analysis, if 0 g y 1 F Y y F X g y g y So, g y u y f Y u This integral has no analytic solution, so we cannot easily invert this result to get g . . However a numerical (or graphical) solution is shown in .

Conversion of uniform pdf to (a) a Gaussian pdf and (b) a Rayleigh pdf. A numerical solution for g X was required for (a) in order to invert , whereas (b) uses the analytic solution for g X , given in . We can get an analytic solution to this problem as follows. If we generate a 2-D Gaussian from polar coordinates, we need to use two random variables to generate r and θ with the correct distributions. In particular, r requires a Rayleigh distribution which can be integrated analytically and hence gives a relatively simple analytic solution for g . . Assuming we start with a uniform pdf from 0 to 1 as before, generating θ is easy as we just scale the variable by 2 to get random phases uniformly distributed from 0 to 2 . Once we have r θ , we can convert to Cartesian components x 1 x 2 to obtain two variables with Gaussian pdfs. To generate r correctly, we need a Rayleigh pdf f R r r r 2 2 r 0 0 So, g y r y f R r r 0 y r r 2 2 0 y r 2 2 1 y 2 2 To get y g x , we just invert the formula for x g y . Hence x 1 y 2 2 y 2 2 1 x x 0 x 1 y g x -2 1 x This conversion is illustrated in . Summarizing the complete algorithm: Generate a 2-D random vector x x 1 x 2 with uniform pdfs from 0 0 to 1 1 , by two calls to a standard random number generator function (e.g. rand() in Matlab; although this whole procedure is unnecessary in Matlab as there is already a Gaussian random generator, randn()). Convert x 1 into r with Rayleigh pdf using r g x 1 -2 1 x 1 Convert x 2 into θ with uniform pdf from 0 to 2 using θ 2 x 2 Generate two independent random variables with Gaussian pdfs of unit variance and zero mean using y 1 r θ and y 2 r θ Repeat steps 1 to 4 for each new pair of variables required. y 1 and y 2 may be scaled by σ to adjust their variance, and an offset may be added in order to produce a non-zero mean.