ADONGO DISTRIBUTION

Consider that the possible outcomes of a given time period or space n can be divided into two categories; those that cause a given event to occurs in a particular time or space and those that do not.

Consider the probability distribution of a time or a space event that occurs randomly but has uniform probability of occurrences and if n denote as time or space and x_i denote the number of times the event occurs at a particular time or space. Then, the probability which follows above characteristics is Adongo probability distribution.

P(x) = [(αx / (x + 1)] [α + (β / x)]^x         x = 1, 2, 3, ……………….

MEAN

Given Adongo distribution of successful outcome in time or space n for x times occurrence of the event. The mean of the distribution is given as;

E(x) = αn

VARIANCE

Given Adongo distribution of successful outcomes α and failure β in time space n for x times occurrences of the event. The variance of the distribution is given as;

Var(x) = αβn

CHARACTERISTICS OF ADONGO DISTRIBUTION

1.    There are exactly two possible outcomes in a given time or space. We can think of these two outcome as “success” and “failure”

2.    The probability of certain number of success during a given interval of times or space is determine.

3.    The individual success are independent of one another

4.    The success will occur uniformly over the entire time or space under consideration.

EXAMPLE

A firm manufactures long rolls of tape and then cuts the rolls into 12-ft lengths

Extensive measurement have shown that defects in the roll occurs at random with probability 0.40

a. What is the probability that a tape has at most 4 defects?

b. What is the probability that a tape has more than 2 defects?

c. If a case contain 3 tapes, what is the probability that one tape has 4 defects and the two remaining tape have 3 defects each?

      

 SOLUTION

       P( x ≤ 4) = [ (0.4 * 1) / (1 + 1)] [0.4 + (0.6 / 1)]¹

                                                      +

                         [(0.4 * 2) / (2 + 1)] [ 0.4 + (0.6 / 2)]²

                                                      +

                          [(0.4 * 3) / (3 + 1)] [0.4 + (0.6 / 3)]³

                                                      +

                         [( 0.4 * 3) / (4 + 1)] [ 0.4 + (0.6 / 4)]⁴

P(x ≤ 4) = 0.2 * 1 + 0.267 * 0.49 + 0.3 * 0.216 + 0.32 * 0.09

P(x ≤ 4) = 0.2 + 0.13 + 0.065 + 0.029

P(x ≤ 4) = 0.424