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© 2018 IJRAR November 2018, Volume 5, Issue 4 www.ijrar.org (E-ISSN 2348-1269, P- ISSN 2349-5138)

IJRAR1BHP080 International Journal of Research and Analytical Reviews (IJRAR) www.ijrar.org 415

The Applications of Discrete Distribution In Daily Life

Situations Rajesh Kumar1 and Kamalpreet kaur2

2Department of Mathematics, School of Chemical Engineering and Physical Sciences, Lovely Professional University, Punjab, India

Abstract: This paper deals with the discrete distributions and their applications. Few research papers have been reviewed

about the discrete distributions and its applications. All these research papers have been formulated into the separate

paragraphs. The discete distributions like Binomial distribution and Poisson distribution have been applied on the data and

observations have been discussed accordingly. For the discrete random variables, the probability can be described with the

discrete distributions like binomial distribution, Poisson, distribution, negative binomial distributions, Bernoulli distribution,

etc. The discrete distributions and their applications in the medical field, biological field, quality analysis in laboratories have

been discussed in the literature review.

Keywords: Discrete distribution, Bernoulli distribution, Quality assurance.

1.Introduction

The discrete distribution is a distribution which deals with the discrete random variable. Binomial distribution,

Poisson distribution, Binomial negative distribution and several others. . A theoretical probability distribution gives

the law in accordance of which the different values of random variable are distributed with specified probabilities

according to some mathematical laws.

Binomial distribution is consumed to determine the quantity of successes in fixed trials. Poisson distribution

happens if there are events that do not originate from a certain number of experimental trials but arise at a random

time and space level. The phenomena where variance is larger than mean like number of insect bites leads to

negative binomial distribution.

2.Literature Review

Shipra Banik and B.M. Golam Kibria [1] in 2009 addressed other discrete models in the data sample and their

contrast in case of high frequency zeros. To achieve this, Goodness of fit statistics was calculated for all discrete

models. It was concluded that in the case of high frequency of zeros in the data sample, Negative binomial, Zero

inflated Poisson, Zero inflated negative binomial models fit well. To illustrate these models, real-life examples have

been used where the negative binomial approach works well in all processes except for the number of patients

attending a hospital every day;this data was fit well by zero truncated negative model.

John Gurland [2] in 1959 investigated that the data obtained in the medical and biological research may be fit well

in negative binomial and other spreadable frequency allocations. The 𝑝. 𝑚. 𝑓 that the negative binomial random

variable 𝑌 assumes for 𝑥 is

P{X=x}=(

1

𝑞)

𝑘𝑘(𝑘+1)(𝑘+𝑥−1)

𝑥!(

𝑝

𝑞)

𝑥

where 𝑥 = {0,1,2 … } (1)

The experimentations done in the medical and biological sciences are related to statistical distributions which may

be normal but may be discrete. The distributions of survival times of patients cured by cancer is a continuous

distribution which is non-normal. The method of frequencies and a common of the two above methods may be used.

Maximum likelihood is an efficient method for estimating the parameters.

K.Teerapabolarn and K. Jaioun [3] in 2014 derived an approximation of binomial distribution by an improved

Poisson distribution with parameters n and p and mean=nλ when n→∞ and p→0, Poisson distribution could be

exhausted as an estimate of a B.D (binomial distribution). This estimate is more precise than the Poisson

approximation while 𝑛 is sufficiently large.

A discrete random variable X has the binomial distribution if the 𝑝𝑚𝑓 is as below:

𝑃(𝑋 = 𝑥) =(𝑛𝑥

)𝑝𝑥𝑞𝑛−𝑥, (2)

𝑥 = 0,1,2 … , 𝑛 where parameters 𝑛 𝜖 ℕ and pϵ (0, 1)

The form |bn,p(x)- pλ(x)| for x={0,1,2…,n} was used for measuring the accuracy of the Poisson approximation.

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Emilio Gómez-Déniz, José Maŕia Sarabia, Enrique Caldeŕin-Ojeda [4] in 2011 a discrete distribution was adopted,

based on two parameters 𝛼 < 1, α ≠0 and 0 < 𝜃 < 1. Geometric distribution is the limiting case of this new

distribution for phenomena such as automobile insurance which have two features: over-dispersion and zero

inflated, a new distribution with cumulative distribution function has been introduced as admires for the random N

variable that takes non-negative integer {0,1,…}

Pn== 1 −ln(1−𝛼𝜃𝑛+1)

ln (1−𝛼) (3)

This distribution proved useful for modeling zero-inflated count data and dispersion count data.

S. B. Hassan at el [5] in 2012 a one factor discrete distribution. Due to limited applications of usual discrete

distributions like negative binomial, poisson for failure times, reliability, etc, many distributions were introduced.

Weibul distribution was introduced having the probability mass function 𝑃{𝑋 = 𝑥) = 𝑞𝑥𝛽

- 𝑞(𝑥+1)𝛽 ∀ 𝑥 ∈ {0,1,2, … } where 𝑞𝜖(0,1), 𝛽 > 0. (4)

In the context of the renewal theory, the Weibul distribution finds applications in testing contagion; pacemaker

processing animal timing: estimation of relic senescence through population dynamics models etc. The 𝑝. 𝑚. 𝑓 of

the new distribution is

p(x)= 𝑝𝑥

1+𝜃{(1-p)(1+θx)+θ(1-2p)}, where 𝑝 = 𝑒−𝜃, ∀ 𝜃 > 0 𝑎𝑛𝑑 𝑥 = {0,1,2, …}

The continuous Lindley distribution is given by the above equation.

Consul P.C. and Jain G.C. [6] in 1973 obtained a new generalization of the λ1 and λ2 parameters of poisson

distribution which is the limiting form of universal negative binomial distribution.

For λ1>0, |λ2|<1, generalized poisson distribution is defined as

pz(λ1,λ2)= λ1(λ1+xλ2)x-1 𝑒

−(𝜆1+𝜆2)

𝑥! , 𝑥 = {0,1,2, …} (5)

s.t 𝑝(λ1,λ2)=0 ∀ 𝑥𝜖(𝑚,∞) 𝑖𝑓 𝜆1 + 𝑚𝜆2 ≤ 0

The value of the variance of this distribution depends on λ2. It may be more than , equal to or less than the mean

accordingly as λ2 is real. This generalization was approximated using James Stirling’s formula. This generalization

fits well to entirely statistics in which average and variance differs through |λ2|≤0.5.

C.F. Linda at el [7] in 1983 applied binomial distribution for the excellence assertion of the quantifiable chemical

studies of the reference sample. The laboratories analyzing many reference sample make use of the binomial

distribution for evaluating laboratory performance. Small laboratories use method of “standard additions” for

quality analyses.Where the number of variance was real except zero, the individual values were considered skewed

and if the individual values exceeded two standard deviations, this indicates lack of precision. The theory of

Extreme runs defined by Geant and Leavenworth had consumed to solve the problem. The B.D (binomial

distribution) used as follow:

P(x)= 2{𝑁!

𝑖!(𝑛−𝑖)!(0.5)𝑖 (0.5)𝑁−𝑖} (6)

𝑊ℎ𝑒𝑟𝑒 𝑝(𝑥)= probability of having at least 𝑥 objects the identical side are on zero axis.

i= number of points on same side of zero line

N= number of successive points

Samuel S. Shapiro and Hassan Zahediin [8] 1990 developed a number of discrete distributions in expressions of

Bernoulli random variables. The applications of Bernoulli was also discussed. Bernoulli trials are the building

blocks of the discrete distributions.

There can be only two outcomes in any one trial. The parameter p is defined as the probability that X=1. Such

random variable is assumed to a Bernoulli distribution by the 𝑝𝑚𝑓

f(x)= px(1-p)1-x where x= 0,1 (7)

For applying a distribution, firstly it is important to confirm if it is a Bernoulli trial or not. For checking “what is

the probability that the first success occurs on Yth trial?”, geometric model is used, if we need to find “what is the

probability of Y successes in n trials?”, binomial distribution is appropriate.




Jerzy Letkowsi [9] in 2012 discussed the appliances of the Poisson distribution in his paper. To title some are- the

variety of alterations on a constituent of DNA per unit time, the numeral of system failures per day, number of

persons visiting a website per minute, etc.

Consider a random variable N(t), such that N(t)=max{n: Sn(t)≤t} and entirely variables Xk, k=1,2,3,… have similar

exponential distribution, f(x)=μ𝑒−𝜇𝑥, x≥0, N(t) has the distribution:

f(t,n)= P{N(t)=n}= 𝑒−𝜇𝑡(𝜇𝑡)𝑛

𝑛!, F(t,n)= P{N(t)≤n}= ∑ 𝑓(𝑡, 𝑘)𝑘=𝑛

𝑘=0 , n=0,1,2,… (8)

Time based Poisson variable is more popular as compared to the space orientated poisson variable. It find

application in counting the number of insects found in a 1-square foot area of farm land, number of eagles nesting in

a domain, etc. The presentation of statistical cases should be enriched by appropriate business, social background

description. This makes “technical” cases more interesting.

Zahoor Ahmad, Adil Rashid and T.R. Jan [10] in 2017 introduced a new discrete compound distribution.This

distribution was attained through compounding size unfair Consul Distribution by generalised beta distribution.The

compounding of probability allows us to obtain both discrete and continuous distributions. A flexibility of

compound distributions was analysed through many research papers. A discrete random variable is declared to have

is its 𝑝. 𝑚. 𝑓. is specified as

P(X=x)= {(

1

𝑥) ( 𝑚𝑥

𝑥−1)𝑝𝑥−1(1 − 𝑝)𝑚𝑥−𝑥+1

0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

, x=1,2,3,… (9)

where 0<p<1 and 1≤m≤p-1

4.Numerical Example

The compound distribution was applied on on bunching traffic in Australian rural highways. The paper proposed a

compound of size biased consul distribution with generalized beta distribution by compounding, the size biased

consul distribution with universal beta distribution.

Table 1: Application of Discrete Distribution to “Estimate the increasing population in India”

S.No. Year (Population in

millions)

B(n,p) Pois(λ)

1 1990 822 0.711109 4.52519E-17

2 1991 839 0.243821 3.83227E-15

3 1992 856 0.040407 2.30016E-13

4 1993 872 0.00431 7.97134E-12

5 1994 892 0.000333 4.4393E-10

6 1995 910 1.98E-05 1.1271E-08

7 1996 928 9.41E-07 2.00475E-07

8 1997 946 3.69E-08 2.51536E-06

9 1998 964 1.21E-09 2.24112E-05

10 1999 983 3.39E-11 0.000156629

11 2000 1001 8.13E-13 0.000704262

12 2001 1019 1.69E-14 0.002291323

13 2002 1040 3.06E-16 0.006075038

14 2003 1056 4.84E-18 0.009603621

15 2004 1072 6.71E-20 0.011914705

16 2005 1089 8.18E-22 0.011535348

17 2006 1106 8.77E-24 0.008565895

18 2007 1122 8.25E-26 0.005098763

19 2008 1138 6.81E-28 0.002416058

20 2009 1154 4.92E-30 0.000914308

21 2010 1170 3.09E-32 0.000277187




The growth of population takes place at a rate of 1.03%.

The mean of the data in the column named years is 1078.933

The population values are in millions so are the values of Binomial and Poisson probability functions.

5. Conclusions:

From the above data calculations and the chart, it can be concluded that year by year, the population in india is

increasing quite rapidly. As such the value of Binomial distribution is decreasing and values of Poisson distribution

is increasing from year 1990 to 2010 and then started decreasing. As here n=30 so Poisson distribution gives better

resuls than the Binomial distribution.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

B(n,p)

Pois(λ)

22 2011 1186 1.68E-34 6.75254E-05

23 2012 1220 7.86E-37 1.64667E-06

24 2013 1235 3.13E-39 2.36369E-07

25 2014 1251 1.04E-41 2.43905E-08

26 2015 1267 2.86E-44 2.05121E-09

27 2016 1283 6.28E-47 1.40955E-10

28 2017 1299 1.06E-49 7.93467E-12

29 2018 1316 1.3E-52 3.00716E-13

Figure 1 showing the variation of Binomial and Poisson distribution with increasing population yearly(from

1990-2018)




References

1. Banik,S.,Kibria,B.M.G.(2009)’On Some Discrete Distributions and their Applications with Real Life

Data’.Journal of Modern Applied Statistical Methods8(2),423-447

2. Gurland,J.(1959)’Some Applications of the Negative Binomial and other contagious

distributions’.Journal of Modern Applied Statistical Methods49(10),1388-1399

3. Teerapabolarn,T.,Jaioun,K.(2014)’Approximation of Binomial distribution by an improved Poisson

distribution’.International Joural of Pure and Applied Mathematics97(4),491-495

4. Déniz,E.G.,Sarabia,J.M.,Ojeda,E.C.(2011)’A new discrete distribution with actuarial

applications’.Insurance-Mathematics and Economics48(3),406-412

5. Bakouch,H.S.,Jazi,M.A.,Nadarajah,S.(2011)’A new discrete distribution’.AJournal of theoretical

and applied statistics48(3),200-240

6. Consul,P.C.,Jain,G.C.(1973)’A Generalisation of the Poisson distribution’.Technometrics15(4),791-

799

7. Friedman,L.C.,Bradford,W.L.,Peart,D.B.(1983)’Application of Binomial Distributions to quality

assurance of quantitative chemical analysis’.Journal of Environmental Sciences and

Health18(4),561-570

8. Shapiro,S.S.,Zahedi,H.(1990)’Bernoulli Trials and Discrete Distributions’.Journal of Quality

Technology22(3),193-205

9. Letkowski,J.(2012)’Developing Poisson distribution applications in a cloud’.Journal of Case

Research in Business and Economics5,1

10. Ahmad,Z.,Rashid,A.,Jan,T.R.(2017)’A New Discrete Compound Distribution with

Applications’.Journal of Statstical Applications and Probability6(1),231-241

11. Gupta,S.C.,Kapoor,V.K.’Fundamentals Of Matematical Statistics’.Sultan Chand and Sons11,8.2-

9.69


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