On Some Probability Results Characterizing the Distribution in
Micro-economic Structures using the Formula of Faà Di Bruno
Ghizlane Chaachoui and Mohammed El Khomssi
Modelling and Scientific Computing Laboratory, Faculty of Science and Technology of Fez,Box 2202,
University S. M. Ben Abdellah Fez, Morocco
Keywords: Micro-Economic Structures, Individuals, Faà Di Bruno’s Formula.
Abstract: In this article we will treat the distribution of individuals in micro-economic structures in two different
cases. In the first one, no constraints on the types are imposed on individuals or structures, in the second
one, three types of constraints are imposed. This work is finished by an application that combines between
two techniques, one purely theoretical based on the formula of Faà Di Bruno and the other is very practical
having an economic aspect, in order to help and to facilitate the management of the distributions of the
individuals in micro-economic structures in the discrete frame.
1 INTRODUCTION
When we have m micro-economic (Avinash, 2014)
structures and we want to engage n individuals in
these projects, we have two cases to consider:
Assignment without constraints: In this case
we will deal with two types of assignment:
Assign the individuals to these projects without
placing constraints on their skills or the type of
micro-economic structures.
All the individuals have the same skills and all
the projects are of the same type.
Assignment with constraints: In this case we
will have three types of assignment:
Random distribution of the individuals.
Distribution of the individuals in an ascending
order.
Distribution of the individuals with
conditioning.
2 COMBINATORIAL RESULTS
LINKING MICRO-ECONOMIC
STRUCTURES AND
INDIVIDUALS
2.1 Possible Surjections
This part is dedicated to finding the number of
possibilities to assign the individuals to the m
projects in the first kind of assignment, i.e. the one
without constraints on the type. The two
assignments above are identical, since the
assignment in both cases is unconstrained.
Each possibility represents a surjection from the
set of all individuals to the set of all micro-economic
structures. By nature the number of individuals is
greater than the number of projects, so we work
under the assumption m n.
We define:
= {
,
,…,
} the set of individuals.
= {
,
,…,
} the set of micro-economic
structures.
,
the number of surjections from
to
.
Calculating the number of possible assignments
of n individuals in m micro-economic structures is
244
Chaachoui, G. and El Khomssi, M.
On Some Probability Results Characterizing the Distribution in Micro-economic Structures using the Formula of Faà Di Bruno.
DOI: 10.5220/0009772302440248
In Proceedings of the 1st International Conference of Computer Science and Renewable Energies (ICCSRE 2018), pages 244-248
ISBN: 978-989-758-431-2
Copyright
c
2020 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
equivalent to calculating
,
the number of
surjections from
to
.
Proposition. The number of surjections
,
from
to
verifies the following equalities:
,
,
(1)
,
,
,
(2)
,
1
(3)
Proof. See (Robert, 2003), (Rowan, 2009) and
(Martin, 2007).
Remark. the terms 1
change signs, but
,
> 0 for all n and m n, indeed
,
=1 and
,
= n !, the other cases are justified by the
principle of recurrence of the decomposition of
,
.
2.2 Possible Partitions
Keep in mind that the set of individuals is
which
contains n elements. To calculate the possible
partitions of this set we introduce the n-th number of
Bell (Jean, 2017) that presents the number of
partitions of a set with n distinct elements, using the
following formula
1
!
(4)
There is a recursive formula to compute this
number step by step which is the formula of Aitken,
and it is represented as following
(5)
There is also a close connection between the
number of possible surjections and the number of
possible partitions, defined as follows:
,
!
(6)
3 COMBINATORICS AND SET OF
STRUCTURES
Assuming that we have n individuals that we
distribute in m microeconomic structures classified
by type
,
,…,
. For exemple we take the poor
as the individuals and the Income Generating
Activities (Shiree, 2011) and (Carletto, 2007) as the
micro-economic structures. For their types we can
have: livestock and animal production, sale of
artisanal or agricultural products…
We define Ω as the universe made up of ways to
place n individuals in m microeconomic structures.
3.1 Random Distribution
When we want to distribute the individuals in the
structures in a random fashion, the way to
differentiate between two structures will be only by
the number of individuals in each one of them.
However there can have zero or more than one
individual in each structure.
In this case the number of ways to allocate our n
individuals in the m microeconomic structures will
be the number of ways to write the integer n as the
sum of m natural integers. It is equal to
Card Ω=
(7)
3.2 Distribution of the Individuals in
an Ascending Order
If we build distributions of n individuals numbered
from 1 to n, the difference between two structures
will be only by the individual’s number that contains
each one of them. It is the most obvious coding of
the individuals; it's all about ordering them, using
the date of engagement for example.
Each individual have m ways to choose their
respective micro-economic structure. So
Card Ω =
(8)
3.3 Distribution of the Individuals with
Conditioning
In this last case, each micro-economic structure can
either be empty or containing one and only one
individual, as a result: n m. Therefore
Card Ω =
(9)
On Some Probability Results Characterizing the Distribution in Micro-economic Structures using the Formula of Faà Di Bruno
245
4 APPLICATION:
DISTRIBUTION IN SEVERAL
REGIONS
4.1 Application 1
Assuming we have q regions
,
,…,
and we
classify the micro-economic structures according to
d types
,
,…,
. We put:
the random variable (Walter, 2008) which
denotes the number of individuals in the
structure of type
in the region
.
.
the random variable that designates the
number of individuals in the structure of type
in all the regions such as
.
∑
(10)
As an example of the type
:
the number of individuals in the structure of
the type
in the region 1.
the number of individuals in the structure of
the type
in the region 2.
.
.
the number of individuals in the structure of
the type t
in the region q.
.
∑
Theorem 1. The probability law (Sharma, 2009)
of the random variable
.
is given by the following
expression:
P[
.
=k]=
∑
∑
∑
.
0
∏
.
!
(11)
Proof. Keep in mind that
,
,…,
are an independent and identical random variables in
. for
.
=
∑
the generating function (Henk, 2012) and (Geoffrey,
2014) in this case can be calculated as :
.
=
.
(12)
If we take a function f(t)=
, we will have
.
.
.
We can also calculate the probability law using
the generating function by using the following
equality (Norman, 2005),
.
0
= k! P[
.
]
(13)
Which means that
.
0
= k! P[
.
]=
.
0
(14)
Then we introduce the formula of Faà Di Bruno
(Johnson, 2002) and (El Khomssi, 2016) that gives
the m-th derivative of a composite function
=
∑
∑
∑
!
∏
!
!
∏
(15)
In our case f(t)=
and g(t)=
.
(t), so we have
.
=
∑
.
∑
∑
!
∏
!
!
∏
.
(16)
We choose to take t=0 in order to apply (14).
Therefore
.
0
=
∑
.
0
!
∏
!!
∑
∑
∏
!
.
=
∑
.
0
!
∏
!
∑
∑
∏
.
=
∑
.
0
∑
∑
!
∏
.
!
And then we divide by k! to have the desired
result.
Theorem 2. When q is also a random variable,
the probability law of the random variable, that we
denote in this case
.
, is given by
ICCSRE 2018 - International Conference of Computer Science and Renewable Energies
246
P[
.
=k]=
∑
∑
∑
.
0
∏
.
!
(17)
Proof. This time q is also a random variable
and
,
,…,
an independent and identical
random variables in the set . We still have
.
=
∑
.
in this case the generating function (Geoffrey,
2014) is defined as
.
= (
.
)(
t
)
(18)
We directly apply the formula of Faà Di Bruno with
f(t)=
(t) and g(t)=
.
(t), so
.
(t)=
∑
.
∑
∑
!
∏
!!
∏
.
(19)
For t=0,
.
(0)=
∑
.
0
∑
∑
!
∏
!!
∏
!
.
=
∑
.
0
∑
∑
!
∏
!
∏
.
Then we divide by k! to find the final expression
of P[
.
=k].
Theorem3. The k-th moment (Geoffrey, 2014)
of the random variable
.
can be written as
.
=
∑
E[
.
=
∑
∑
!
!
∑
∑
!
∏
!!
∏
.
(20)
And for the random variable
.
we have
.
=
∑
E[
.
=
∑
∑
∑
∑
1
!
∏
!!
∏
.
(21)
In order to prove both cases, we calculate the
high order moment
.
and
.
of the
two random variables by following exactly the same
steps of the two proofs above (depending on the
case) using this time the link between the generating
function and the high order moment (Norman, 2005)
presented as
E[
.
]=
.
1
(22)
And using the formula that link the high order
moment to the factorial moment (we use the Stirling
number of the second kind properties (Weisstein,
2002)
.
=
∑
E[
.
(23)
4.2 Application 2
In this application we still have q regions
,
,…,
and we classify the micro-economic
structures according to d types
,
,…,
. We also
put :
the random variable (Walter, 2008) which
denotes the number of individuals in the
structure of type
in the region
.
But this time we have:
S
.
the random variable that designates the
number of individuals in the region R
in all
the structures, such as
.
∑
(24)
As an example of the region
:
the number of individuals in the structure of
the type
in the region 1.
the number of individuals in the structure of
the type
in the region 1.
.
.
the number of individuals in the structure of
the type t
in the region 1.
.
∑
On Some Probability Results Characterizing the Distribution in Micro-economic Structures using the Formula of Faà Di Bruno
247
The same results of the application 1 can be
obtained by following exactly the same calculation
procedure with
,
,…,
an independent
and identical random variables in .
This time we will have:
.
=
.
(25)
When d is also a random variable we will have:
.
= (
.
)(
t
)
(26)
5 CONCLUSIONS
As a conclusion, we have treated a method that
simplifies the calculations and helps to facilitate the
management of the distributions of the individuals in
the micro-economic structures in the discrete frame.
A study of the continuous case with the proper
application will be the object of the next paper.
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