The concept of transform appear naturally for investigation of different problems in mathematics, physics, and engineering sciences. The main reason to introduce them is that they greatly simplify the calculations. The type of transformation depends on the problem itself, that is why varieties of transform occur, see for example, Z-transform, moment generating function, Laplace-transform, Fourier-transform, Mellin-transform, Hankel-transform, etc. Moreover, they may have different names as well, e.g., probability generating function, characteristic function. This chapter is devoted to the probability generating function and the Laplace-transform which are closely related to the discrete and continuous nonnegative random variables. Their usefulness will be illustrated by several examples.
Of course, there many books dealing with special transform, but in this material I concentrate on our needs only keeping in mind their applications in queueing theory. As basic sources I recommend the following books: Allen [ 2 ], Kleinrock [ 48 ], Trivedi [ 94 ].
Definition 3.1.
Let
be a nonnegative discrete random variable having
distribution 
, 
.
Then the generating
function
of 
is defined as
is defined if the series is
convergent.
Theorem 3.2. The generating function holds the following properties
1. 
,
2. 
,
3. 
,
4. 
,
5. 
, 
.
.2. 
.
3. 
.
4. 
Collecting the terms we get
Theorem 3.3.
If
are independent then 
Proof. In the proof we use the theorem if the random variables are independent then the mean of their product is equal to the product of their means. Thus we can write
Theorem 3.4. Generating function of a random sum
Proof. By the law of total expectation we get
The following two theorems play an important role in many applications and simplify the calculations in limiting distributions.
Theorem 3.5.
(Continuity Theorem)
Let
be a sequence of nonnegative, integer valued
random variables. If the sequence of the corresponding distribution
converges to a distribution, that is 
and 
, where 
then the corresponding generating functions of
converge to the generating function of
at any point in 
, that is
where
If the limit 
exists, but 
, then the convergence of the generating functions
holds only in 
.
Remark 3.6. For illustration let us see the following example
Let 
, that is 
, and 
, if 
, then
However
Theorem 3.7.
(Continuity Theorem)
If a sequence of
the generating function of 
converges to a function 
on 
, then the sequence of the corresponding
distribution of 
converges to a probability distribution with
generating function 
.
Remark 3.8.
If
we assume that 
exists but only on 
, then 
is not necessarily a generating function as we see
in the following example.
Example 3.1. If
a random variable 
takes the values 
and 
with the same probability then 
and thus 
. It is easy to see that 
is not valid since
Generating Function of Important Distributions
Example 3.2. Find the generating function of a Bernoulli distribution, then the mean and variance.
Solution:
Example 3.3. Find
the generating function of a Poisson distribution with parameter
, and then the mean and variance.
Solution:
Example 3.4. By using the generating function approach find the convolution of two Poisson distributions.
Solution:
Applying the properties of the generating function it is easy to get the generating function of the sum and by taking into account the generating function of a Poisson distribution we have
that exactly the generating function of a Poisson distribution
with parameter 
.
Example 3.5. By
the help the generating functions show that 
if 
, 
such that 
!
Solution:
Use that if 
then 
!
We are going to show that the generating function of the binomial distribution converges to the generating function of the Poisson distribution, that is
that is the generating function of a Poisson distribution with
parameter 
.
Example 3.6. Let
and independent, 
. Find the generating function of the random sum
.
Solution:
By using the relationship to the generating function of a random sum and taking into account the form of the generating function of a Bernoulli and Poisson distribution after substitution we shall get the desired result. Therefore we can write
which shows that 
.
Example 3.7. Solve the following system of differential equations
Solution:
Multiplying both sides of the equations by the appropriate power
of 
we get
Let us introduce the generating function 
as
By adding both sides we have
The initial condition is
Thus the system of differential equations reduces to a single differential equation, namely
with initial condition
Rearranging the terms we get
and the solution is
Since 
and 
, thus 
that is 
.
Hence 
which shows that 
is the generating function of a Poisson
distribution with parameter 
. Therefore the solution of the system of
differential equation is
