Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 8s (2024) 

 

565 
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On Discrete Harmonic Distributions related with Harmonic Mean 

Random Variable 

 

Prasanta Kumar Das 

School of Applied Sciences, KIIT  Deemed to be  University,  Bhubaneswar-751024 (India) 

pkdasfma@kiit.ac.in /  dasprasantkumar@yahoo.co.in 

 

Article History: 

Received: 20-05-2024 

Revised: 03-07-2024 

Accepted: 21-07-2024 

Abstract:  

The main purpose of this paper is to introduce the concept of harmonically distributed 

discrete random variable. We define probability mass function such as harmonic mass 

function, exponential harmonic mass function, natural logarithmic  harmonic mass 

function and their associated cumulative distribution functions. Existence the 

distribution is shown with some examples. Finally harmonic mass function is used to 

solve a run-time problem and a capacitance problem as an application of the 

distribution  in the field of electrical engineering. 

Keywords: Harmonic mean random variable,  Harmonic mass function, Exponential 

harmonic mass function,  Natural logarithmic  harmonic mass function. 

AMS 2010 Subject Classification: 62Exx 

 

1. Introduction and Preliminary 

In the theory of probability and statistics, the continuous harmonic distribution or harmonic law was 

studied by Etienne Halphen (1941) as a special case of generalized inverse Gaussian distribution 

family with 𝛾 = 0. The Geometry of statistical notions or the connections between Geometry and 

Statistics has existed from the beginning of both disciplines (Adler C. F. 1958 [1], and Fisher, J. B, 

1978 [4] ). The connection between arithmetic, geometric, and harmonic mean can be studied by the 

same approach of Geometry of Statistics. You may observe that some other investigations are based 

on the statistical properties of geometric notions (Ahangar R. R. 2010 [2], Hilbert, D. 1902 [5], 

Pearson, R. 2011 [10], Saville, J. D., and Wood R. G., (1977) [11]. Arithmetic mean as a great mode 

of investigation in statistics is used in many natural phenomena, but many researches prefer to use 

either geometric or harmonic mean in their investigations, (MacCluer, C. R. 2000 [7]). The 

arithmetic mean and the median may be the most popular and convenient measures that financial 

analysts use for their valuations. It is believed by many financial organizations and bankers that the 

Harmonic Mean provides better information for reasonable measure in investment strategy (Mathews 

and Gilbert- 2006, Meyer D., 1970 [8]). In slowly-decaying distributions, the harmonic mean often 

turns out to be a much better characterization than the arithmetic mean, which is a reciprocal 

transformation generally not even well-defined theoretically for these distributions (Pearson, R. 2011 

[10]). Much research has been done and computational tools designed these days that are equipped to 

change the mode of computation in either arithmetic, geometric, or harmonic sense. The continuous 

harmonic density function, transformation (horizontal shift) of harmonic density function, harmonic 

density with stretch or contraction, and general harmonic function are studied by (Ahangar R. 2013 



Communications on Applied Nonlinear Analysis 

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[3]. Later he developed the algorithm of continuous harmonic modelling and continuous harmonic 

regression. 

The total capacitor (𝐶𝑇) in an series circuit of 𝑛 capacitance 𝐶𝑖, 𝑖 = 1,2,⋯ , 𝑛 has a harmonic relation 

and the total resistance (𝑅𝑇) in an parallel circuit of 𝑛 resistance 𝑅𝑖 , 𝑖 = 1,2,⋯ , 𝑛 has a harmonic 

relation defined by 

circuits 𝑅𝑇 𝐶𝑇 

Series  

𝑅1 + 𝑅2 +⋯+ 𝑅𝑛 
(
1

𝐶1
+
1

𝐶2
+⋯+

1

𝐶𝑛
)
−1

 

Parallel 
(
1

𝑅1
+
1

𝑅2
+⋯+

1

𝑅𝑛
)
−1

 
 

𝐶1 + 𝐶2 +⋯+ 𝐶𝑛 

In order to reduce the voltage drop in the series circuit, we need to reduce the 𝐶𝑇 and 𝑅𝑇 in parallel 

circuit. According to recent study the drops 𝐶𝑇 or 𝑅𝑇 in the line is admissible upto 5% either in series 

or parallel respectively. Therefore a problem is there to find the probability of joining of (𝑘 + 1)𝑡ℎ 

capacitor after 𝑘𝑡ℎ capacitors in the series line shouldn’t exceed 0.05. This concept encourages us to 

develop the discrete harmonic distribution. 

To the best of the author’s knowledge, no direct investigation exists to introduce discrete harmonic 

probability mass functions, moments, central moments and its applications. In this paper we develop 

the random variable for some types of discrete harmonic distribution and their cumulative 

distribution mass functions.  

2. Harmonic Mean RV (HMRV) and Discrete Harmonic Distribution 

For any real (𝑛 + 1) real values 𝑎0, 𝑎1, ⋯ , 𝑎𝑛, harmonic convexity of 𝐴 = {𝑎0, 𝑎1, ⋯ , 𝑎𝑛} is  

𝐻𝐶(𝐴) = [∑
𝑖=0

𝑛

(𝛼𝑖/𝑎𝑖)]
−1

=
𝑎0𝑎1⋯𝑎𝑛

∑
𝑖=0

𝑛
𝛼𝑖𝑎0𝑎1⋯𝑎𝑖−1𝑎𝑖+1⋯𝑎𝑛

  

satisfying ∑
𝑖=0

𝑛

𝛼𝑖 = 1. Taking 𝛼𝑖 =
1

𝑛+1
, we have 

𝐻𝐶(𝐴) = [∑

𝑖=0

𝑛

(𝛼𝑖/𝑎𝑖)]

−1

= [∑

𝑖=0

𝑛

(
1

(𝑛 + 1)𝑎𝑖
]

−1

= (𝑛 + 1)
𝑎0𝑎1⋯𝑎𝑛

∑
𝑖=0

𝑛

𝑎0𝑎1⋯𝑎𝑖−1𝑎𝑖+1⋯𝑎𝑛

. 

Let 𝑋 be the harmonic mean random variable consists of successive harmonic mean values of 𝑎𝑖’s, 

i.e., 𝑋 = {𝑥0, 𝑥1, ⋯ , 𝑥𝑛} where  

𝑥𝑘 = [∑

𝑖=0

𝑘

(1/𝑎𝑖)]

−1

=
𝑎0𝑎1⋯𝑎𝑘

∑
𝑖=0

𝑘

𝑎0𝑎1⋯𝑎𝑖−1𝑎𝑖+1⋯𝑎𝑘

                                        (2.1) 

 for 𝑘 = 0,1,⋯ , 𝑛, then we have  



Communications on Applied Nonlinear Analysis 

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𝑥0 = 𝑎0, 𝑥1 =
𝑎0𝑎1
𝑎0 + 𝑎1

, 𝑥2 =
𝑎0𝑎1𝑎2

𝑎1𝑎2 + 𝑎0𝑎2 + 𝑎0𝑎1
, ⋯ , 𝑥𝑛

=
𝑎0𝑎1⋯𝑎𝑛

𝑎1𝑎2⋯𝑎𝑛 + 𝑎0𝑎2⋯𝑎𝑛 + 𝑎0𝑎2⋯𝑎𝑛−1
. 

Consider 𝑝(𝑥) =
𝐶

𝑥
 such that ∑

𝑖=0

𝑛

𝑝(𝑥𝑖) = 1 for some 𝐶 > 0. Assume that  

𝐹(𝑥0) = 𝑝(𝑥0) =
𝐶

𝑎0
 

𝐹(𝑥1) = 𝑝(𝑥0) + 𝑝(𝑥1) = 𝐶 [
1

𝑎0
+
𝑎0 + 𝑎1
𝑎0𝑎1

] = 𝐶
𝑎0 + 2𝑎1
𝑎0𝑎1

= 𝐶 [
1

𝑎1
+
2

𝑎0
] 

𝐹(𝑥2) = 𝑝(𝑥0) + 𝑝(𝑥1) + 𝑝(𝑥2) 

= 𝐶 [
1

𝑎0
+
𝑎0 + 𝑎1
𝑎0𝑎1

+
𝑎1𝑎2 + 𝑎0𝑎2 + 𝑎0𝑎1

𝑎0𝑎1𝑎2
] 

= 𝐶
𝑎0𝑎1 + 2𝑎0𝑎2 + 3𝑎1𝑎2

𝑎0𝑎1𝑎2
= 𝐶 [

1

𝑎2
+
2

𝑎1
+
3

𝑎0
]. 

 In general, for 𝑘 = 0,1,⋯ , 𝑛, we have  

𝐹(𝑥𝑘) = 𝐶 [
1

𝑎𝑘
+

2

𝑎𝑘−1
+

3

𝑎𝑘−2
+⋯+

𝑘 − 1

𝑎2
+
𝑘

𝑎1
+
𝑘 + 1

𝑎0
] = 𝐶∑

𝑘

𝑖=0

𝑖 + 1

𝑎𝑘−𝑖
. 

Now  

1 = 𝑝(𝑥0) + 𝑝(𝑥1) + 𝑝(𝑥2) + ⋯+ 𝑝(𝑥𝑛) 

= 𝐶 [
1

𝑎0
+
𝑎0 + 𝑎1
𝑎0𝑎1

+
𝑎1𝑎2⋯𝑎𝑛 + 𝑎0𝑎2⋯𝑎𝑛 + 𝑎0𝑎2⋯𝑎𝑛−1

𝑎0𝑎1⋯𝑎𝑛
] 

= 𝐶 [
1

𝑎𝑛
+

2

𝑎𝑛−1
+

3

𝑎𝑛−2
+⋯+

𝑛 − 1

𝑎2
+
𝑛

𝑎1
+
𝑛 + 1

𝑎0
] = 𝐶∑

𝑛

𝑖=0

𝑖 + 1

𝑎𝑛−𝑖
 

⇒ 𝐶 = (∑

𝑖=0

𝑛
𝑖 + 1

𝑎𝑛−𝑖
)

−1

                                                              (2.2). 

 Hence 

𝑝(𝑥) = (∑

𝑖=0

𝑛
𝑖 + 1

𝑎𝑛−𝑖
)

−1
1

𝑥
, 𝑥 = 𝑥0, 𝑥1, ⋯ , 𝑥𝑛.                   (2.3) 

 is a special type of harmonic pmf in the harmonic set.  

Example 2.1 If a runner runs as per the time (in seconds) given in the time space 𝑇 = {𝑡0, 𝑡1, ⋯ , 𝑡𝑛}, 

𝑡𝑖 ≠ 0 with relative speed has a harmonic random variable 𝑋𝑇 = {𝑥: 𝑥 = 𝑥(𝑡), 𝑡 ∈ 𝑇} =

{𝑥0, 𝑥1, ⋯ , 𝑥𝑛} obtained by equation (2.1). The runner has a relative distance covered in time 𝑡 ∈ 𝐴 



Communications on Applied Nonlinear Analysis 

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has the harmonic pmf measured by (2.3). Thus for 𝑎 = 2,3,5, the equation (2.1) gives harmonic rv 

𝑋𝑇 with images 𝑥 = 2,
6

5
,
30

31
. Since equation (2.2) gives 𝐶 =

30

71
, we have the harmonic pmf 𝑝(𝑥) =

30

71𝑥
, 𝑥 = 2,

6

5
,
30

31
 obtained by (2.3) satisfies ∑

𝑖=0

𝑛

𝑝(𝑥𝑖) = 1. Hence the random variable 𝑋𝑇 has the 

pmf values are  

𝑝(2) =
15

71
, 𝑝(6/5) =

25

71
, and 𝑝(30/31) =

31

71
 

and cdf values are  

𝐹(2) =
15

71
, 𝐹(6/5) =

40

71
, and 𝐹(30/31) = 1. 

 Example 2.2 In a series with three capacitors 𝐶1 = 2𝜇F, 𝐶2 = 4𝜇F and 𝐶3 = 5𝜇F are connected, 

then rv 𝑋𝐶 measures total capacitance after connecting to the next capacitor, then 𝑋𝐶 = {𝑥1, 𝑥2, 𝑥3} 

where 𝑥1 = 2, 𝑥2 = (
1

2
+
1

4
)
−1

= 4/3 and 𝑥3 = (
1

2
+
1

4
+
1

5
)
−1

= 20/19. The harmonic pmf of 𝑋𝐶 is 

𝑝(𝑥) =
5

11𝑥
  for 𝑥 = 2, 4/3, 20/19, i.e., 𝑝(2) =

5

22
, 𝑝(4/3) = 15/44 and 𝑝(20/19) = 19/44 and 

the cdf of 𝑋𝐶 are 𝐹(2) =
5

22
, 𝐹(4/3) = 25/44 and 𝐹(20/19) = 1.  

3. Discrete Exponential Harmonic Distribution 

 Let ℤ𝑚
𝑛  be the set of integers lie in between 𝑚 and 𝑛 for 0 ≤ 𝑚 ≤ 𝑛 − 1 and 𝐾 = {𝑥 ≠ 0: 𝑥 =

𝑘 exp (
1

𝑘−1
) , 𝑘 ≥ 1} satisfying 𝑥𝑘.  

Let 𝑋 be a discrete random variable defined by 𝑋 = {𝑧𝑥 ∈ ℝ: 𝑥 = 0,1,⋯ , 𝑛} and the discrete 

harmonic probability mass function (dhpmf) be  

𝑝(𝑥; 𝑧) =
𝐶

(𝑛 + 1)𝑧𝑥
, 𝑥 = 0,1,⋯ , 𝑛, and, 0 otherwise, 

for some 𝐶 > 0 satisfying 0 ≤ 𝑝(𝑥; 𝑧) ≤ 1 and ∑𝑛𝑥=0 𝑝(𝑥; 𝑎) = 1. Since 0 ≠ 𝑥 ∈ 𝐾, we obtain 𝑥 =

𝑘exp (
1

𝑘−1
), i.e., 𝑥𝑘 = 𝑘𝑥 for some 𝑘 > 0 and 𝑥 > 0. 

Letting 𝑧𝑥 = 𝑎
𝑥 for some parameter 𝑎 > 0, 𝑥 = 0,1,2,⋯ , 𝑛, we obtain  

𝑝(𝑥; 𝑎, 𝑛) =
𝐶

(𝑛 + 1)𝑎𝑥
, 𝑥 = 0,1,⋯ , 𝑛, and, 0 otherwise, 

Considering it as a legitimate pmf, we have unity property, i.e.,  

1 =∑

𝑛

𝑥=0

𝑝(𝑥; 𝑎, 𝑛) =∑

𝑛

𝑥=0

𝐶

(𝑛 + 1)𝑎𝑥
 

=
𝐶

(𝑛 + 1)
(1 +

1

𝑎
+
1

𝑎2
+⋯+

1

𝑎𝑛−1
+
1

𝑎𝑛
) =

𝐶𝑎−𝑛

(𝑛 + 1)
(1 + 𝑎 + 𝑎2 +⋯+ 𝑎𝑛) 



Communications on Applied Nonlinear Analysis 

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 implying 𝐶 = (𝑛 + 1) (
𝑎−1

𝑎−𝑎−𝑛
) for 𝑎 > 1. Thus fixed fixed parameters 𝑛 = 2,3,⋯ and 𝑎 = 2,3,⋯, 

the dhpmf is obtained as  

𝑑ℎ(𝑥; 𝑎, 𝑛) = 𝑃(𝑋 = 𝑥) = (
𝑎 − 1

𝑎 − 𝑎−𝑛
)
1

𝑎𝑥
, 𝑥 = 0,1,⋯ , 𝑛;   𝑎𝑛𝑑 0 otherwise 

and the corresponding discrete harmonic probability distribution function (DHPDF) is given by  

𝐷𝐻(𝑥; 𝑎, 𝑛) = 𝑃(𝑋 ≤ 𝑥) =∑

𝑛

𝑥=0

𝑑ℎ(𝑦; 𝑎) = (
𝑎 − 1

𝑎 − 𝑎−𝑛
)∑

𝑥

𝑦=0

1

𝑎𝑦
, 𝑥 ≤ 𝑛 

= (
𝑎 − 1

𝑎 − 𝑎−𝑛
) 𝑎−𝑥 (

𝑎𝑥+1 − 1

𝑎 − 1
) =

𝑎 − 𝑎−𝑥

𝑎 − 𝑎−𝑛
  for  𝑥 = 0,1,⋯ , 𝑛. 

We define the discrete complementary harmonic probability distribution function (DCHPDF) as  

𝐶𝐻(𝑥; 𝑎, 𝑛) = 𝑃(𝑋 ≥ 𝑥) = 1 − 𝑃(𝑋 ≤ 𝑥 − 1) = 1 −
𝑎 − 𝑎−(𝑥−1)

𝑎 − 𝑎−𝑛
=
𝑎−(𝑥−1) − 𝑎−𝑛

𝑎 − 𝑎−𝑛
 

for 𝑥 = 0,1,⋯ , 𝑛. Thus for 𝑎 = 2, the dhpmf is  

𝑑ℎ(𝑥; 𝑎) = (
1

2 − 2−𝑛
)
1

2𝑥
, 𝑥 = 0,1,2,⋯𝑛;  and 0 otherwise 

and the DHPDF is  

𝐷𝐻(𝑥; 2, 𝑛) =
2 − 2−𝑥

2 − 2−𝑛
, 𝑥 = 0,1, 2,⋯ , 𝑛. 

For 𝑎 = 3, the dhpmf is  

𝑑ℎ(𝑥; 3,3) = (
2

3 − 3−𝑛
)
1

3𝑥
, 𝑥 = 0,1,2,⋯ , 𝑛;  and 0 otherwise 

and the DHPDF is  

𝐷𝐻(𝑥; 3, 𝑛) =
3 − 3−𝑥

3 − 3−𝑛
. 𝑥 = 0,1,2,⋯ , 𝑛. 

The accompanying table is given for 𝑎 = 2 and 𝑛 = 3:  

𝑎 = 2, 𝑛 = 3 

 𝑥 = 0 𝑥 = 1 𝑥 = 2 𝑥 = 3 

𝑑ℎ(𝑥; 2,3) 0.5333 0.2667 0.1333 0.0667 

𝐷𝐻(𝑥; 2,3) 0.5333 0.8000 0.9333 1.0000 

𝑎 = 2, 𝑛 = 3 

 𝑥 = 0 𝑥 = 1 𝑥 = 2 𝑥 = 3 

𝑑ℎ(𝑥; 3,3) 0.6750 0.2250 0.0750 0.0250 

𝐷𝐻(𝑥; 3,3) 0.6750 0.9000 0.9750 1.0000 

 



Communications on Applied Nonlinear Analysis 

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The accompanying table is given for 𝑎 = 𝑒 and 𝑛 = 1,2,3:  

𝑎 = 𝑒, 𝑛 = 1 

 𝑥 = 0 𝑥 = 1   

𝑑ℎ(𝑥; 𝑒, 1) 0.7311 0.2689   

𝐷𝐻(𝑥; 𝑒, 1) 0.7311 1.0000   

𝑎 = 𝑒, 𝑛 = 2 

 𝑥 = 0 𝑥 = 1 𝑥 = 2  

𝑑ℎ(𝑥; 𝑒, 2) 0.6652 0.2447 0.0009  

𝐷𝐻(𝑥; 𝑒, 2) 0.6652 0.9100 1.0000  

𝑎 = 𝑒, 𝑛 = 3 

 𝑥 = 0 𝑥 = 1 𝑥 = 2 𝑥 = 3 

𝑑ℎ(𝑥; 𝑒, 3) 0.6439 0.2369 0.0871 0.0321 

𝐷𝐻(𝑥; 𝑒, 3) 0.6439 0.8808 0.9679 1.0000 

4. Discrete Natural Logarithmic Harmonic Distribution 

For any real 𝑥 > 1, we have ln𝑥 > 0. Let 𝑎0 ≥ 𝑎𝑘 for 𝑘 = 1,2,⋯ , 𝑛 such that the greatest integer of 

𝑎0 is at least 𝑛 + 1. Let 𝑋 be the harmonic mean random variable with images 𝑥𝑘, 𝑘 = 0,1,2,⋯ , 𝑛 

obtained by the relation given in (2.1). We have  

𝑥0 = 𝑎0, 𝑥1 =
𝑎0𝑎1
𝑎0 + 𝑎1

, 𝑥2 =
𝑎0𝑎1𝑎2

𝑎1𝑎2 + 𝑎0𝑎2 + 𝑎0𝑎1
, ⋯,  

𝑥𝑛 =
𝑎0𝑎1⋯𝑎𝑛

𝑎1𝑎2⋯𝑎𝑛 + 𝑎0𝑎2⋯𝑎𝑛 + 𝑎0𝑎2⋯𝑎𝑛−1
, 

so 𝑥𝑘 > 1 for each 𝑘 as [𝑎0] ≥ 𝑛 + 1. The logarithmic harmonic pmf of 𝑋 is defined by  

𝑝(𝑥; 𝑛) = {

𝐶

(𝑛 + 1) ln 𝑥
,  if  𝑥 = 𝑥0, 𝑥1, ⋯ , 𝑥𝑛; 

0,  otherwise,
  

 

for some 𝐶 > 0 satisfying 0 ≤ 𝑝(𝑥; 𝑛) ≤ 1 and ∑
𝑥=0

𝑛

𝑝(𝑥; 𝑛) = 1 where  

ln 𝑥𝑘 = ln

(

 
𝑎0𝑎1⋯𝑎𝑛

(∑
𝑖=0

𝑘

𝑎0𝑎1⋯𝑎𝑖−1𝑎𝑖+1⋯𝑎𝑘))

 =∑

𝑘

𝑖=0

ln 𝑎𝑖 − ln (∑

𝑖=0

𝑘

𝑎0𝑎1⋯𝑎𝑖−1𝑎𝑖+1⋯𝑎𝑘). 

for for 𝑘 = 0,1,⋯ , 𝑛.  

1 = ∑

𝑛

𝑘=0

𝑝(𝑦𝑘; 𝑛) =
𝐶

(𝑛 + 1)
∑

𝑛

𝑘=0

1

ln 𝑥𝑘
 

=
𝐶

(𝑛 + 1)
[
1

ln 𝑥0
+

1

ln 𝑥1
+

1

ln 𝑥2
+⋯+

1

ln 𝑥𝑛−1
+

1

ln 𝑥𝑛
] 



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⇒ 𝐶 = (𝑛 + 1) [
1

ln 𝑥0
+

1

ln 𝑥1
+

1

ln 𝑥2
+⋯+

1

ln 𝑥𝑛−1
+

1

ln 𝑥𝑛
]
−1

. 

 Thus the logarithmic harmonic pmf of 𝑋 is  

𝑝(𝑥) = 𝑙ℎ(𝑥; 𝑛) = {

𝑘

ln 𝑥
,  𝑖𝑓  𝑥 = 𝑥0, 𝑥1, ⋯ , 𝑥𝑛; 

0,  otherwise,
  

 

where  

𝑘 = [
1

ln 𝑥0
+

1

ln 𝑥1
+

1

ln 𝑥2
+⋯+

1

ln 𝑥𝑛−1
+

1

ln 𝑥𝑛
]
−1

, 

ln 𝑥0 = ln 𝑎0 

ln 𝑥1 = ln 𝑎0 + ln𝑎1 − ln(𝑎0 + 𝑎1) 

ln 𝑥2 = ln 𝑎0 + ln 𝑎1 + ln 𝑎2 − ln(𝑎0𝑎1 + 𝑎0𝑎2 + 𝑎1𝑎2) 

 and so on,  

ln 𝑥𝑛 =∑

𝑛

𝑘=0

ln 𝑎𝑘 − ln(∑

𝑘=0

𝑛

𝑎0𝑎1⋯𝑎𝑘−1𝑎𝑘+1⋯𝑎𝑛). 

provided 𝑥𝑘 > 1 for each 𝑘 = 0,1,2⋯ and cdf of 𝑋 is 𝐿𝐻(𝑥; 𝑛) = 𝑃(𝑋 ≤ 𝑥) for all 𝑥 =

𝑥0, 𝑥1, ⋯ , 𝑥𝑛 given by  

𝐿𝐻(𝑥0; 𝑛) = 𝑝(𝑥0; 𝑛) =
𝑘

ln 𝑥0
,  

𝐿𝐻(𝑥1; 𝑛) = 𝑝(𝑥0; 𝑛) + 𝑝(𝑥1; 𝑛) = 𝑘 [
1

ln 𝑥0
+

1

ln 𝑥1
] 

𝐿𝐻(𝑥2; 𝑛) = 𝑝(𝑥0; 𝑛) + 𝑝(𝑥1; 𝑛) + 𝑝(𝑥2; 𝑛) = 𝑘 [
1

ln 𝑥0
+

1

ln 𝑥1
+

1

ln 𝑥2
] 

𝐿𝐻(𝑥3; 𝑛) = 𝑝(𝑥0; 𝑛) + 𝑝(𝑥1; 𝑛) + 𝑝(𝑥2; 𝑛) + 𝑝(𝑥3; 𝑛) 

= 𝑘 [
1

ln 𝑥0
+

1

ln 𝑥1
+

1

ln 𝑥2
+

1

ln 𝑥3
]   and so on. 

 Example 4.1 Let 𝑛 = 4. For 𝑎0 = 5, 𝑎1 = 5.1, 𝑎2 = 5.2, 𝑎3 = 5.3, 𝑎4 = 5.4, we have 𝑥0 = 5, 𝑥1 =

2.52475, 𝑥2 = 1.69956, 𝑥3 = 1.28689, 𝑥4 = 1.03923. Therefore  

𝑙ℎ(𝑥; 4) = {

𝑘

ln 𝑥
,  𝑖𝑓  𝑥 = 𝑥0, 𝑥1, ⋯ , 𝑥𝑛; 

0,  otherwise,
  

 

where  



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

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𝑘 = [
1

ln 𝑥0
+

1

ln 𝑥1
+

1

ln 𝑥2
+

1

ln 𝑥3
+

1

ln 𝑥4
]
−1

= 0.0298 

which is approximated to 4 significant digit form. Thus  

 ln 𝑥0 = 1.60944; ln 𝑥1 = 0.92614; ln 𝑥2 = 0.53037; ln 𝑥3 = 0.25223; ln 𝑥4 = 0.03848, 

implying the pmf of 𝑋 is 

𝑝(𝑥0; 5) =
𝑘

ln 𝑥0
= 0.01852; 𝑝(𝑥1; 5) =

𝑘

ln 𝑥1
= 0.03218; 𝑝(𝑥2; 5) =

𝑘

ln 𝑥2
= 0.05619 

𝑝(𝑥3; 5) =
𝑘

ln 𝑥3
= 0.11815; 𝑝(𝑥4; 5) =

𝑘

ln 𝑥4
= 0.77443. 

 and cdf of 𝑋 is 𝐿𝐻(𝑥; 𝑛) = 𝑃(𝑋 ≤ 𝑥) for all 𝑥 = 𝑥0, 𝑥1, ⋯ , 𝑥𝑛 given as follows.  

𝐿𝐻(𝑥0; 4) = 𝑝(𝑥0; 4) =
𝑘

𝑙𝑛𝑥0
= 0.01853, 

𝐿𝐻(𝑥1; 4) = 𝑝(𝑥0; 4) + 𝑝(𝑥1; 4) = 𝑘 [
1

ln 𝑥0
+

1

ln 𝑥1
] = 0.05073 

𝐿𝐻(𝑥2; 4) = 𝑝(𝑥0; 4) + 𝑝(𝑥1; 4) + 𝑝(𝑥2; 4) = 𝑘 [
1

ln 𝑥0
+

1

ln 𝑥1
+

1

ln 𝑥2
] = 0.10695 

𝐿𝐻(𝑥3; 4) = 𝑝(𝑥0; 4) + 𝑝(𝑥1; 4) + 𝑝(𝑥2; 4) + 𝑝(𝑥3; 4) 

= 𝑘 [
1

ln 𝑥0
+

1

ln 𝑥1
+

1

ln 𝑥2
+

1

ln 𝑥3
] = 0.22518 

𝐿𝐻(𝑥4; 4) = 𝑝(𝑥0; 4) + 𝑝(𝑥1; 4) + 𝑝(𝑥2; 4) + 𝑝(𝑥3; 4) + 𝑝(𝑥4; 4) 

= 𝑘 [
1

ln 𝑥0
+

1

ln 𝑥1
+

1

ln 𝑥2
+

1

ln 𝑥3
+

1

ln 𝑥4
] = 1.00012. 

 The distribution value at 𝑥 ≥ 𝑥4 is approximated to 1 − ϵ where ϵ = 0.00055 because of 

truncational errors present in 𝑥𝑘’s and pmf values in 𝑥𝑘’s.  

Future Scope 

 Our next aim to study the mean, variance, moment, skewness, Kurtosis, moment generating 

functions of the discrete harmonic mean distributed random variables with the harmonic pmf, 

exponential harmonic pmf, natural logarithmic harmonic pmf in future.   

Refrences 

[1] Adler, C. F., (Claire Fisher Adler) 1958, “Modern Geometry", An Integrated First Course, Second Edition, C.W. 

Post College of Long Island University, McGraw-Hill Book Company. 

[2]  Ahangar, R. R., 2010, “Introduction to Statistical Geometry", Proceedings of the International Conference on 

Scientific Computing, Editors: Hamid R. Arabnia, George A. Gravvanis, CSREA Press, www.world-academy-of-

Science.org,SCS. 

[3] Ahangar, R., (2013), “Harmonic Probability Density Function, Modelling and Regression", European International 

Journal of Science and Technology, 2(5) 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 8s (2024) 

 

573 
https://internationalpubls.com 

[4] Fisher, B. J. (Box, John Fisher), “ R. A. Fisher, A life of Scientist", New York, Wiley, 1978. 

[5] Hilbert, D., (1902), "The Foundations of Geometry", Chicago, IL: The Open Court Publishing Company. 

[6] Kapur, D., and Mundy, J. L., (1989), Geometric Reasoning", A Bradford Book. A collection of papers presented at 

international workshop on geometric reasoning held at Oxford University on 1986. 

[7] MacCluer, C. R. (2000), "Industrial Mathematics, Modeling in Industry, Science, and Government", Prentice Hall. 

[8] Mathews and Gilbert, (2006) "When Averaging Multiples, Apply the Harmonic Mean", BVUpdate: A Business 

Valuation Library Publication, www.BVLibrary.com. 

[9] Meyer D., (1970), "Probability and Statistics an Undergraduate Course", W.A. Benjamin INC. New York. 

[10] Pearson, R., (November 11, 2011), "Harmonic means, reciprocals, and ratios of random Variables", this article was 

first published on ExploringDataBlogs, and later contributed to R-bloggers: www.r- bloggers.com. 

[11] Saville J. D., and Wood, R. G., Third print (1997), "Statistical Methods: The Geometric Approach, Springer. 

[12] Tesi di, L. D., (2006-2007), "Simulation and Analysis of Chemical Reactions using Stochastic Differential 

Equations", Thesis, Universita Degli Studi Di Torino, Anno Academico.