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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.
© 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 416
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.
© 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 417
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
© 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 418
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)
© 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 419
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