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.