Quotients of Uniformly Distributed Random Variables

In many cases random variables having absolutely continuous distributions can be represented as quotients of uniformly distributed random variables.

- Combined with acceptance-rejection sampling (see Section 3.2.3) this yields another type of simulation algorithm.
- The mathematical foundation for this type of algorithm is the following transformation theorem for the density of absolutely continuous random vectors (that has already been mentioned e.g. in Section 1.2.3 of the course ,,Statistik I'').

- Let be an absolutely continuous random vector with joint density and let be a Borel-measurable function with continuous partial derivatives .
- Let now the Borel-set
be picked in a way such
that
- Let be the inverse of .
- Then the random vector
is also absolutely
continuous and the density
of
is given by

which is the same as

From Theorem 3.8 we obtain the following result concerning the representation of absolutely continuous random variables as quotients of uniformly distributed random variables.

- Let
be Borel measurable and bounded such
that

- Let the random vector be uniformly distributed on the
(bounded) Borel set

- Then the quotient is an absolutely continuous random
variable with density
where

**Proof**-
- Notice that (27) implies that the Borel set
defined in (28) is bounded, i.e.
.
This is due to the following reasons.
- For the inequality is equivalent to .
- If on the other hand it is equivalent to .
- Therefore

and

- The following joint density
of the random
vector is thus well defined
- The function
where
and
- is a bijection of onto itself
- and its functional determinant is given by

- Theorem 3.8 therefore implies
- that the density
of the random vector
has the following form:
- Moreover, the marginal density
of the second
component von
is given by

- that the density
of the random vector
has the following form:

- Notice that (27) implies that the Borel set
defined in (28) is bounded, i.e.
.
This is due to the following reasons.
**Example**-
*(normal distribution)*- Theorem 3.9 yields a third method to generate N-distributed pseudo-random numbers (as an alternative to the Box-Muller algorithm from Section 3.2.1 and the polar method explained in Section 3.2.3).
- Consider the function
where
for all
. For the bounds in
(29) we obtain:
- According to Theorem 3.9 a sequence
of N-distributed pseudo-random numbers can now be
generated as follows.
- 1.
- Generate a -uniformly distributed pseudo-random number and a -uniformly distributed pseudo-random number .
- 2.
- If , i.e., if , then return to step 1.
- 3.
- Otherwise put .