1. Introduction
In industry, training programmes are conducted with the aim of training new workers to do particular job repeatedly every day. It is assumed that a particular trainee will show progress proportional to the number of days he attends the program, otherwise his ability will be different from one day to another, see [1] [4] .
Let
be the length of a programme in days and
the number of repetitions of the job per day a trainee has to do. If a trainee is responding to the instructions, it would be reasonable to assume the probability that he will do a single job right, i.e. the probability of success on the
day is
, see Abdulnasser and Khidr [1] ,
and hence the probability that he will do
jobs correctly out of
jobs on the
day is
,
and
.
When a trainee is not responding to the instructions,
will be a constant
,
To test whether a trainee is responding or not, we test if
is varying or sustaining a constant value
This can be done by computing the total number of jobs that have been done correctly over the whole period of the program.
Let
stand for the number of jobs done correctly out of
jobs on
day, ![]()
and
,
. In case
,
, the distribution of
will be
.
In this article, we study a generalization of Bernoulli learning model based on probability of success
where
positive integer,
are real numbers,
and
and
is positive integer. This gives the previous results given in [1] - [3] as special cases, where
and
respectively. In Section 2, the probability function
of this model and some properties of the model are obtained. In Section 3, we derive the limiting distribution of the model. Finally, in Section 4, we discuss some special cases.
2. The Generalized Bernoulli Learning Model
Theorem 1. The distribution function of
is
(1)
where
.
Proof. To derive the distribution of Bernoulli learning model based on the sum of the independent random variable
,
where the probability of success is
we define the event
as the event
,
see [5] , and the sum
![]()
where
the generalized Stirling number of the first kind (Comtet numbers), defined by Comtet in [6] [7] as follows
![]()
where
, for more details, see [8] and [9] .
Employing the inclusion-exclusion principle, see [5] , we get
![]()
then
![]()
hence
![]()
![]()
this yields (1). ![]()
Lemma 1.
(2)
(3)
Proof. Consider the pair of inverse relation, see [10]
(4)
Then using (1), let
![]()
Hence from (4), we get
(5)
and setting
, we have
(6)
But we have, see [7]
(7)
Thus
and this yields (2).
If putting
in (5), we get
![]()
using (7), we have
, then
![]()
hence
![]()
![]()
this yields (3). ![]()
3. Limiting Distribution of the Bernoulli Learning Model
In this section we study the limiting distribution of the Bernoulli learning model based on the probability with success ![]()
Theorem 2. Let
where
and
are independent random variables. Then
where
i.e.
is
as ![]()
Proof. The moment generating function of
is
![]()
and the moment generating function of
is
![]()
![]()
![]()
therefore, we have
![]()
by using (2) and (3), we obtain
![]()
(8)
which is the moment generating function of standard normal distribution
![]()
4. Some Special Cases
In this section we discuss some special cases as follows.
i) Setting the probability of successes
we have the results derived in [1] , as special case
Theorem 3. The distribution of
is given by [1]
(9)
where
are the usual stirling numbers of the first kind, see [10] .
Also, they obtained the limiting distribution of learning model, mean and variance as follows.
Theorem 4. Let
where
and
’s are independent random variables. Then
where
i.e.
has
as ![]()
Lemma 2.
(10)
ii) Setting the probability of successes
we have the results derived in [2] , as special case
Theorem 5. The distribution of
is given by [2]
(11)
Lemma 3.
![]()
iii) Setting the probability of successes
we have the results derived in [3] , as special case
Theorem 6.
(12)
where
,
and
p-Stirling numbers, see [11] [12] .
Theorem 7. Let
where
and
are independent random variables. Then
where
i.e.
has
as ![]()
Lemma 4.
![]()
![]()
5. Conclusion
Our main goal of this work is concerned with studying the extension of generalized Bernoulli learning model with probability of success
and
is positive integer. Some previous results, see [1] - [3] , are concluded as special cases of our result, that is for
and
respectively. The mean and variance of the model are obtained. Finally, the limiting distri- bution of the general model is derived. This model has many applications in industry, specially for training pro- grammes.