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Finite Range Probability Distribution for Reliability Theory and 

Theoretical Physics 

 

Musallam 𝐓𝐚𝐛𝐨𝐨𝐤𝟏, Masood 𝐀𝐥𝐚𝐦𝟐, Sabir Ali 𝐒𝐢𝐝𝐝𝐢𝐪𝐮𝐢𝟑, Shradha 𝐃𝐰𝐢𝐯𝐞𝐝𝐢𝟒, Sameen 

Ahmed 𝐊𝐡𝐚𝐧𝟓 and Awdhesh 𝐏𝐚𝐧𝐝𝐞𝐲𝟔 

1,3,5: Department of Mathematics and Sciences, Dhofar University, Salalah, Sultanate of Oman, 

2: Department of Mathematics &IT Center for Preparatory Studies, Sultan Qaboos University, Muscat, Oman. 

4: University of Technology and Applied Sciences – Salalah, Sultanate of Oman. 

6: Department of Applied Sciences(Maths), Ananad Engineering College, Agra, India. 

 

Article History: 

Received: 12-01-2025 

Revised: 15-02-2025 

Accepted: 01-03-2025 

Abstract:  

The aim of this paper is to develop a probability distribution which can be used in the 

reliability theory and in theoretical Physics. Some basic properties of the distribution 

have been discussed and their mathematical forms have been evaluated. 

Keywords: Finite Range distribution 

 

1. INTRODUCTION: 

Development of new probability distributions definitely increases the family of probability 

distributions and reduces the risk of using approximately near distribution. There are many research 

workers who have developed probability distribution in recent times, such as, Mukherjee and Islam 

(1983), Siddiqui et al(1992,1994,1995,2016). 

This distribution will be useful when observations are presented in the form of percent increment or in 

decreasement.  

Probability is the language of statistical mechanics.  It is also fundamental to the understanding of 

quantum mechanics.  In statistical mechanics the physical problems concern large groups of particles, 

like molecules in a gas.  It is not possible to track every single particle’s motion; statistical mechanics 

uses probability distributions to describe the average behavior of the system.  Probability is the 

backbone of thermodynamics.  It enables us understand the likelihood of a system transitioning 

between different states and how it evolves over time.  Unlike classical mechanics, in quantum 

mechanics particles do not have fixed properties.  Their behavior is described by probability 

distributions.  The probability amplitude of finding a particle in a certain state gives us the likelihood 

of that outcome.   

In a radioactive decay process, unstable nuclei transform into more stable ones by emitting particles.  

The Poisson distribution enables us to model the number of decays that could happen in a given time 

period, given the average rate of decay. Reif (2009) discussed Statistical Physics in detail, Roe (2012) 

discussed the role of the theory of probability in experimental Physics. Kuzemsky (2016) discussed 



Communications on Applied Nonlinear Analysis 

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the use of the theory of probability, Michael (2021) discussed the probability related ideas across the 

theory of Physics. 

2. Proposed probability distribution 

The probability density function of the proposed distribution is; 

𝑓(𝑥) =
𝑝

𝑒𝑝−1
𝑒𝑥𝑝  ,   0 < 𝑥 ≤ 1, 𝑝 < 1                 … (1) 

Graph of pdf of the proposed distribution 

Where ‘p’ is the parameter of the distribution, both variable and the parameter are having similar range. 

 

And the cumulative distribution function is; 

𝐹(𝑥) =
𝑒𝑥𝑝−1

𝑒𝑝−1
, 0 < 𝑥 ≤ 1,   𝑝 < 1      … (2) 

 

Graph of cumulative distribution function 

 

Reliability function 

𝑅(𝑡) = 1 − 𝐹(𝑡) = 1 −
𝑒𝑡𝑝 − 1

𝑒𝑝 − 1
=

𝑒𝑝 − 𝑒𝑡𝑝

𝑒𝑝 − 1
,    𝑡 > 0 

p 0.9

p 1.5

p 3

p 10

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

x

P
D

F

p 0.9

p 1.5

p 3

p 10

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

x

C
D

F



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Graph of the Reliability function 

 

Hazard Rate Function 

ℎ(𝑡) =
𝑓(𝑡)

𝑅(𝑡)
=

𝑝𝑒𝑡𝑝 − 1
𝑒𝑝 − 1
𝑒𝑡𝑝 − 1
𝑒𝑝 − 1

=
𝑝𝑒𝑡𝑝

𝑒𝑡𝑝 − 1
 𝑡 > 0 

Graph of the HAZARD RATE function 

 

 

p 0.9

p 1.5

p 3

p 10

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

t

R
t

p 0.9

p 1.5

p 3

p 10

0.0 0.2 0.4 0.6 0.8 1.0

0

5

10

15

20

x

h
x



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3. CHARACTERIZATION OF THE DISITRIBUTION: MOMENTS GENERATING 

AND CHARACTERISTIC FUNCTIONS 

3.1 Moments Generating Function 

    𝑀𝑥(𝑡) = 𝐸(𝑒𝑡𝑥)  

  = ∫ 𝑒𝑡𝑥𝑓(𝑥)𝑑𝑥
1

0
 

            𝑀𝑥(𝑡) =
𝑝(𝑒𝑡+𝑝−1)

(𝑒𝑝−1)(𝑡+𝑝)
   … (3) 

3.2 Characteristic Function 

 

The characteristic function (c.f.) of the model can be obtained as below:  

  𝜙𝑥(𝑡) = 𝐸(𝑒𝑖𝑡𝑥)  

  = ∫ 𝑒𝑖𝑡𝑥𝑓(𝑥)𝑑𝑥
1

0
 

            𝜙𝑥(𝑡) =
𝑝(𝑒𝑖𝑡+𝑝−1)

(𝑒𝑝−1)(𝑖𝑡+𝑝)
    … (4) 

 

4. BASIC PARAMETERS OF THE DISTRIBUTION 

4.1. r th  Moment About Origin  

 

The r t h  moment about origin is given by  

     𝜇𝑟
′ = 𝐸(𝑋𝑟) 

 𝜇𝑟
′ =

Γ(r+1,−p)−Γ(r+1,0)

(−p)r(ep−1)
                             … (5) 

This in turn gives the following results:   

4.2 Mean 

𝑬(𝑿) =  ∫ 𝒙 𝒇(𝒙)𝒅𝒙
𝟏

𝟎

 

𝑬(𝑿) =  ∫ 𝒙
𝒑

𝒆𝒑 − 𝟏
𝒆𝒙𝒑𝒅𝒙

𝟏

𝟎

 

𝑬(𝑿) =  
(𝒆𝒑(𝒑−𝟏)+𝟏)

𝒑(𝒆𝒑−𝟏)
           ….(6)  

 
  

𝝁𝟏
′ = 𝑬(𝑿) =

𝒆𝟐𝒑

𝒆𝒑 − 𝟏
 



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𝝁𝟐
′ = 𝑬(𝑿) =

𝟐𝒆𝟐𝒑

𝒑𝟐(𝒆𝒑 − 𝟏)
 

4.3 Variance 

 

Since variance,  2
2

22 )'1
'())(()()(  −=−= XEXEXV  

 𝑽(𝑿) =
2(𝑒𝑝−1)−𝑝2

𝑝2(𝑒𝑝−1)2     …..(7)   

4.4 Median 

 

To obtain the median we proceed as follows  

 =
−

−
Me

xd
p

xp

0
2
1)1(  

 
2

11
=

−p
Me  

Taking Log on both sides; we get 

 
2

1
lnln)1( =− Mep   

 
p

e
Me

−

−
=

1

010.3
    … (8)   

5. ESTIMATION OF PARAMETER  

6.1 Maximum Likelihood Estimator of Parameter  

L=∏
𝑝.𝑒𝑥𝑝

𝑒𝑝−1

𝑛
𝑖=1  

=
𝑝𝑛

(𝑒𝑝−1)
𝑒𝑝 ∑ 𝑥𝑖

𝑛
𝑖=1  

𝒍𝒏 𝑳 = 𝒏 𝒍𝒏 𝒑 − 𝒏 𝒍𝒏(𝒆𝒑 − 𝟏) + 𝒑 ∑ 𝒙𝒊

𝒏

𝒊=𝟏

 

�̂� =
𝒏(𝒆𝒑 − 𝟏)

𝒏𝒆𝒑 + (𝒆𝒑 − 𝟏 ) ∑ 𝒙𝒊
𝒏
𝒊=𝟏

 

            …..(9) 

ACKNOWLEDGMENT 

 Authors are grateful for the cooperation of all colleagues during the writing of this paper. 



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https://doi.org/10.1080/03057267.2020.1757244   

 

https://doi.org/10.1007/978-1-4684-9296-5
https://doi.org/10.1007/s10773-015-2779-8
https://doi.org/10.1080/03057267.2020.1757244