Fundamental THEOREM for sampling.-

Consider the probability distribution
, i.e.,

= probability for to be between and .

Assume also, for generality, that is limited to the interval (, ).

Then, the THEOREM says that the cummulative normalized distribution is a
uniform distribution:

See Figure in Appendix of Green (reference at bottom), which illustrates
the equipartition of areas implict in eq. (1)

We don't prove this result, which for some people can be considered
"manifestly true."

We verify in eq. (1) that for
, and for
, that is, distribution (1) satisfy the limits corresponding to the
variable .

The THEOREM tell us that the desired value is obtained by solving
("inverting") eq. (1) for .

A) We next discuss some cases of interest that can be solved
analytically, i.e., can be expressed explicitily as a function of .

1) The distribution is uniform in the interval, for instance, the case of the azimuthal angle in scattering.

2) Polar angle for Rutherford scattering for small angles.

Note that

a) The multiplicative factor in the distribution is not relevant for the
sampling.

b) For the case , the distribution reduces to the uniforme
distribution, as it should be.

3) Mean free path (or, also, radiactive or particle decay). These cases correspond to an exponential distribution :

4) Isotropic angular distribution.-
, that
is, we would like to send, on the average, the same number of particles
into each solid angle .

We have expressed here the element of solid angle as the product
of two independent variables, each one with a uniform distribution. For
and
, we get, from (1):

where y are two random numbers uniformly distributed in the unit interval (0,1).

B) If the distribution to be sampled does not have an inverse, that is, if
can not be (easily) expressed as a function of , it is always possible
to use a method of trial and error that eventually gives us a value of
with the distribution we are looking for. This computational method is
known as the "acceptance/rejection method". See, for instance, the
Appendix to Green or the Particle Data Book.

REFERENCES

Dan Green, "The Physics of Particle Detectors", Cambridge University
Press, 2000. See Appendix K, "Monte Carlo models", p. 348.

Particle Data Group, "Review of Particle Physics", Phys. Rev. D,
66,01001-1 (2002). See "32. Monte Carlo Techniques"