Inversion Method

- The following property of the generalized inverse can be used as a basis for the generation of pseudo-random numbers that can be regarded as realizations of random variables whose distribution function is an arbitrary monotonically nondecreasing and right-continuous function such that and .
- Recall the following auxiliary result.
- Let
be an arbitrary distribution function. Then the
function
where

is called the*generalized inverse*of the distribution function . - For arbitrary
and

see Lemma WR-4.1.

- Let
be an arbitrary distribution function. Then the
function
where

- Let be a sequence of independent and uniformly distributed random variables on and let be a distribution function.
- Then the random variables where for are independent and their distribution function is given by .

**Proof**-
- The independence of is an immediate consequence of the transformation theorem for independent random variables; see Theorem WR-3.18.
- Furthermore, (10) implies for arbitrary
and

**Examples**-
- In the following we discuss some examples illustrating
- how Theorem 3.4 can be used in order to generate pseudo-random numbers
- that can be regarded as realizations of independent random variables with a given distribution function .

- These numbers are also referred to as -distributed
pseudo-random numbers
,
- in spite of the fact that the empirical distribution function of the sample
- is only an approximation of for large .

- Note that Theorem 3.4 can only be applied
*directly*if- the generalized inverse of is given explicitly (i.e. by an analytical formula).
- Unfortunately, this situation is merely an exception.

- In the following we discuss some examples illustrating

*Exponential distribution*- Let and
be the distribution function
of the Exp-distribution, i.e.
- Then
for all .

- By Theorem 3.4,
- we have
Exp if and
hence also are uniformly distributed on
- and the pseudo-random numbers
where
- if are realizations of independent and uniformly on distributed random variables .

- we have
Exp if and
hence also are uniformly distributed on

- Let and
be the distribution function
of the Exp-distribution, i.e.
*Erlang distribution*- Let ,
and let
be the
distribution function of the Erlang distribution, i.e., of the
-distribution where

- Then the generalized inverse of cannot be determined
explicitly and therefore Theorem 3.4 cannot be
applied
*directly*.

- However, in Section 1.3.1 of the course ,,Statistik I'' we showed
that
if the random variables
are independent and Exp-distributed.

- By Theorem 3.4
- the pseudo-random numbers
where
- if are realizations of independent and uniformly distributed random variables on .
- In particular, for the pseudo-random numbers can be regarded as realizations of a -distributed random variable.

- the pseudo-random numbers
where

- Let ,
and let
be the
distribution function of the Erlang distribution, i.e., of the
-distribution where
*Normal distribution*- In order to generate normally distributed pseudo-random numbers
one can apply the so-called
*Box-Muller algorithm*, which also requires exponentially distributed pseudo-random numbers. - Assume the random numbers to be independent and uniformly
distributed on .
- By Theorem 3.4, we get that is an Exp-distributed random variable and
- the random vector where
- as for arbitrary

where the last but one equality follows from the substitution

- The pseudo-random numbers
where

- can thus be regarded as realizations of independent and N-distributed random variables ,
- if are realizations of independent and uniformly on distributed random variables .

- For arbitrary
and
the pseudo-random
numbers
where
can be regarded as realizations of
independent and N
-distributed random variables.

- Remarks

- In order to generate normally distributed pseudo-random numbers
one can apply the so-called