Next: Simulation Methods Based on Up: Transformation of Uniformly Distributed Previous: Acceptance-Rejection Method   Contents

### 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'').

Theorem 3.8
• 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

and

which ensures that the restriction of to the set is a bijection where denotes the image of .
• Let be the inverse of .
• Then the random vector is also absolutely continuous and the density of is given by

 (25)

which is the same as

 (26)

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

Theorem 3.9
• Let be Borel measurable and bounded such that

 and (27)

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

 (28)

• 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

 (29)

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

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.