93 

 

 

  

 

 

An Estimation of Survival and Hazard Rate Functions of Exponential 

Rayleigh Distribution 

 

                                                                                                             
 

 

 

Abstract 

      In this paper, we used the maximum likelihood estimation method to find the estimation 

values for survival and hazard rate functions of the Exponential Rayleigh distribution based 

on a sample of the real data for lung cancer and stomach cancer obtained from the Iraqi 

Ministry of Health and Environment, Department of Medical City, Tumor Teaching 

Hospital, depending on patients' diagnosis records and number of days the patient remains in 

the hospital until his death. 

Keyword: Exponential Rayleigh 𝐸𝑅 distribution; Maximum Likelihood Estimation; Chi 

Square Test; Real - Life Data Application. 

1.Introduction   

 Statistical tests are essential for scientific research. Some of these tests are called 

parametric tests and others are called non-parametric tests. These tests help the researcher 

analyze the data accurately and thus are useful in summarizing the results an essentially and 

appropriately way and drawing general conclusions. In parametric tests, some assumptions 

are based the suitability of data for a normal distribution, while non-parametric tests do not 

require that your data follow the normal distribution. 

 Non-parametric tests include various tests. In this study we will use the chi-square 

test which is a non-parametric test and it represents the oldest known goodness of fit test 

introduced by Karl Pearson in )1900(. This test is used to how likely the observed 

Article history: Received   27, June, 2021, Accepted 29, August, 2021, Published in October 2021. 

 

Ibn Al Haitham Journal for Pure and Applied Science 

Journal homepage: http://jih.uobaghdad.edu.iq/index.php/j/index 

 

Lamyaa Khalid 

Hussein 
Department of Mathematics, 

College of Science, 

Mustansiriyah University  

lamyaakhalid8242@gmail

com. 
 

Doi: 10.30526/34.4.2706 

Huda Abdulla Rasheed 
Department of Mathematics, 

College of Science, 

Mustansiriyah University  

iraqalnoor1@gmail.com 

 

Iden Hasan Hussein 
Department of Mathematics, 

College of Science for Women,  
University of Baghdad,Iraq.  

Idenalkanani58@gmail.com 

 

mailto:lamyaakhalid8242@gmail.com
mailto:lamyaakhalid8242@gmail.com
mailto:Idenalkanani58@gmail.com


Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(4)2021 
 

94 

 

distribution of data fits with the distribution that is expected (i.e., to test the goodness of fit), 

it is used to analyze categorical data.  

The goodness of fit test is used to check whether a given sample of data follows from a 

proposed distribution. The formula for calculating a chi-square statistic is [4,6]: 

𝜒2 = ∑
(𝑂𝑖−𝐸𝑖)

2

𝐸𝑖

𝑘
𝑖=1                                                      

Where 

𝑘 represents the number of classes. 

𝑂𝑖 represents the observed frequency in class i 

𝐸𝑖 represents the expected frequency in class i  

The characteristics of the chi-square test is that it can be easily calculated and applied to both 

continuous and discrete variables. It is not recommended for small sample size ( less than 

25). Also the asymptotic distribution of the test  statistic is  the chi-square distribution with 

(𝑘-1) degrees of freedom. The null and alternative hypotheses are as follows: 

𝐻0: The failure time data is distributed as Exponential Rayleigh p.d.f. 

𝐻1: The failure time data is not distributed as Exponential Rayleigh p.d.f. 

The null hypothesis is rejected when comparing the tabulated value at level of significance 

(𝛼) and degree of freedom (𝑘 − 1) is less than the calculated value 𝜒2 > 𝜒2
𝛼,(𝑘 −1)

. 

The exponential distribution is the most commonly used distribution for lifetime data 

analysis. Its simplicity and mathematical feasibility made it the most widely used lifetime 

model in reliability (survival) theory. It is commonly used to model the time until something 

occurs in the process. 

A continuous non-negative random variable 𝑍 is called to have an Exponential distribution 

with parameter 𝛼, if its probability density function is given by [3]: 

𝑓(𝑧; 𝛼)𝐸 = 𝛼 𝑒
−𝛼𝑧                               ; 𝑧 ≥ 0;  𝛼 > 0                      …(1) 

And zero otherwise, where 𝛼 is the scale parameter. 

 

Figure 1: probability density function of Exponential distribution for different values of ( 𝛼 =

0.1, 0.2, 0.3, 0.4,0.5, 0.6, 0.7 ) [Matlab R2015a]. 



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(4)2021 
 

95 

 

The cumulative distribution function is: 

𝐹(𝑧; 𝛼)𝐸 = 1 −  𝑒
−𝛼𝑧                       ; 𝑧 ≥ 0;  𝛼 > 0                       (2)  

 

Figure 2: The cumulative distribution function of Exponential distribution for different values of ( 𝛼 =

0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7 ) [Matlab R2015a]. 

Rayleigh distribution with one parameter is one of the most widely used distributions. It is 

an essential distribution in statistics and operation research. In (1880) Lord Rayleigh 

presented the Rayleigh distribution [5]. It shows the main role in connection with a problem 

in various branches modeling and analyzing lifetime data for instance project effort loading 

modeling, communication, survival and reliability theory, physical sciences, technology, 

diagnostic imaging, clinical subjects, and applied statistics [7]. 

A continuous non-negative random variable 𝑌 is called to have a Rayleigh distribution with 

parameter 𝜆, if its probability density function is given by [2]: 

𝑓(𝑦; 𝜆)𝑅 = 𝜆𝑦 𝑒
− 

𝜆

2
 𝑦2

                               ; 𝑦 ≥ 0;  𝜆 > 0                    (3) 

And zero otherwise, where 𝜆 is the scale parameter. 

 

Figure 3: The probability density function of Rayleigh distribution for different values of ( 𝜆 =

0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7 ) [Matlab R2015a]. 

 The cumulative distribution function is: 

𝐹(𝑦; 𝜆)𝑅 = 1 −  𝑒
− 

𝜆

2
 𝑦2

            ; 𝑦 ≥ 0;  𝜆 > 0                                  (4) 



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96 

 

 

Figure 4: The cumulative distribution function of Rayleigh distribution for different values of ( 𝜆 =

0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7 ) [Matlab R2015a]. 

The Exponential Rayleigh distribution is obtained based on mixed between cumulative 

distribution function of Exponential distribution in equation (2) and cumulative distribution 

function of Rayleigh distributions in equation (4) as follows [1]:        

 Let 𝑋 = 𝑚𝑎𝑥(𝑍, 𝑌) where 𝑍 and 𝑌 are two independent random variables then: 

𝐹(𝑥; 𝛼, 𝜆 )𝐸𝑅 = 𝑝𝑟(𝑋 ≤ 𝑥)        

𝐹(𝑥; 𝛼, 𝜆 )𝐸𝑅 = 1 − 𝑝𝑟(𝑋 > 𝑥)        

𝐹(𝑥; 𝛼, 𝜆 )𝐸𝑅 = 1 − 𝑝𝑟(𝑚𝑎𝑥(𝑍, 𝑌) > 𝑥)        

𝐹(𝑥; 𝛼, 𝜆 )𝐸𝑅 = 1 − [𝑝𝑟(𝑍 > 𝑥). 𝑝𝑟(𝑌 > 𝑥)]       

𝐹(𝑥; 𝛼, 𝜆 )𝐸𝑅 = 1 − [( ∫ 𝑓(𝑧; 𝛼)𝑑𝑧

∞

𝑥

). (∫ 𝑓(𝑦; 𝜆)𝑑𝑦

∞

𝑥

)] 

𝐹(𝑥; 𝛼, 𝜆 )𝐸𝑅 = 1 − [( ∫ 𝛼 𝑒
−𝛼𝑧 𝑑𝑧

∞

𝑥

). (∫ 𝜆𝑦 𝑒
− 

𝜆
2
 𝑦2

𝑑𝑦

∞

𝑥

)] 

𝐹(𝑥; 𝛼, 𝜆 )𝐸𝑅 = 1 −  𝑒
− (𝛼𝑥 + 

𝜆

2
 𝑥2)

            ; 𝑥 ≥ 0; 𝛼, 𝜆 > 0                      (5) 

 

Figure 5: The cumulative distribution function of 𝐸𝑅 distribution for 𝛼 = 0.5 and different values of ( 𝜆 =

0.1, 0.2, 0.3, 0.4,  0.5, 0.6, 0.7 ) [Matlab R2015a]. 

The probability density function is given by [1]: 

𝑓(𝑥; 𝛼, 𝜆 )𝐸𝑅 = (𝛼 + 𝜆𝑥) 𝑒
− (𝛼𝑥 + 

𝜆

2
 𝑥2)

         ; 𝑥 ≥ 0; 𝛼, 𝜆 > 0                        (6) 

And zero otherwise, where 𝛼 𝑎nd 𝜆 are scale parameters. 



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(4)2021 
 

97 

 

 

Figure 6: probability density function of 𝐸𝑅 distribution for 𝛼 = 0.5 and different values of ( 𝜆 =

0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7 ) [Matlab R2015a]. 

The Survival function can be expressed by [1]: 

𝑆(𝑡; 𝛼, 𝜆 )𝐸𝑅 =  𝑒
− (𝛼𝑡 + 

𝜆

2
 𝑡2)

        𝑡 ≥ 0; 𝛼, 𝜆 > 0                                         (7) 

 

Figure 7: Survival function of 𝐸𝑅 distribution for 𝛼 = 0.5 and different values of ( 𝜆 =

0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7 ) [Matlab R2015a]. 

The Hazard rate function is obtained by [1]: 

ℎ(𝑡; 𝛼, 𝜆 )𝐸𝑅 = 𝛼 + 𝜆𝑡               𝑡 ≥ 0; 𝛼, 𝜆 > 0                                         (8) 

 

Figure 8: Hazard rate function of 𝐸𝑅 distribution for 𝛼 = 0.5 and different values of ( 𝜆 =

0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7 ) [Matlab R2015a]. 

The 𝑟𝑡ℎ moment about the origin can be expressed by [1]: 

𝐸(𝑋𝑟)𝐸𝑅 = ∑
(−𝛼)𝑛

𝑛!
∞
𝑛=0

 2 
𝑟+𝑛

2

 𝜆 
𝑟+𝑛

2

[ 
𝛼 

√2𝜆
 𝛤 ( 

𝑟+𝑛+1

2
 ) +  𝛤 ( 

𝑟+𝑛+2

2
 )]   ; 𝑟 = 1,2,3,          (9) 

The mean: The first moment which is named the mean is obtain by put 𝑟 = 1 in equation (9) 

thus [1]: 

𝐸(𝑋)𝐸𝑅 = ∑
(−𝛼)𝑛

𝑛!
∞
𝑛=0

 2 
1+𝑛

2

 𝜆 
1+𝑛

2

[ 
𝛼 

√2𝜆
 𝛤 ( 

𝑛+2

2
 ) +  𝛤 (

𝑛+3

2
 )]                                            (10) 



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(4)2021 
 

98 

 

The variance: The general formula 𝑣(𝑋) of 𝐸𝑅 distribution is given by [1]: 

 𝑣(𝑋)𝐸𝑅 = 𝐸(𝑋
2)𝐸𝑅 − [𝐸(𝑋)𝐸𝑅]

2  

=  [∑
(−𝛼)𝑛

𝑛!
∞
𝑛=0

 2 
2+𝑛

2

 𝜆 
2+𝑛

2

[ 
𝛼 

√2𝜆
 𝛤 ( 

𝑛+3

2
 ) +  𝛤 ( 

𝑛+4

2
 )]] – [∑

(−𝛼)𝑛

𝑛!
∞
𝑛=0

 2 
1+𝑛

2

 𝜆 
1+𝑛

2

[ 
𝛼 

√2𝜆
 𝛤 ( 

𝑛+2

2
 ) +

        𝛤 ( 
𝑛+3

2
 )]]

2

                                                                                                                    (11)                                      

The moment generating function of 𝐸𝑅 distribution can be obtained by [1]:  

𝑀𝑋(𝑡)𝐸𝑅 = ∑
(−(𝛼−𝑡))𝑛

𝑛!
∞
𝑛=0

 2 
𝑛
2

 𝜆 
𝑛
2

[ 
𝛼 

√2𝜆
 𝛤 ( 

𝑛+1

2
 ) +  𝛤 ( 

𝑛+2

2
 )]                                   (12) 

2. Maximum Likelihood Estimation 

           Let 𝑥 = (𝑥1, 𝑥2, … , 𝑥𝑛) be a random sample of size (𝑛) drawn from 𝐸𝑅 distribution 

with pdf given by equation (6). The complete data likelihood function 𝐿(𝛼, 𝜆| 𝑥 )𝐸𝑅
𝑀𝐿𝐸 for a 

given random sample can be expressed by, 

𝐿(𝛼, 𝜆| 𝑥 )𝐸𝑅
𝑀𝐿𝐸 = ∏ 𝑓(𝑥𝑖;

𝑛
𝑖=1 𝛼, 𝜆)𝐸𝑅 = ∏ [(𝛼 + 𝜆𝑥𝑖)𝑒

−(𝛼𝑥𝑖 + 
𝜆

2
 𝑥𝑖

2)
]𝑛𝑖=1                  (13) 

 The natural log – likelihood function is: 

ℓ𝐸𝑅
𝑀𝐿𝐸 = ln 𝐿(𝛼, 𝜆| 𝑥 )𝐸𝑅

𝑀𝐿𝐸 = ∑ ln(𝛼 + 𝜆𝑥𝑖) − ∑ (𝛼𝑥𝑖 +
𝜆

2
𝑥𝑖

2)𝑛𝑖=1
𝑛
𝑖=1                       (14) 

          We derive the natural log – likelihood function partially with respect to 𝛼 and 𝜆 

respectively and setting it equal to zero yields, 

𝜕ℓ𝐸𝑅
𝑀𝐿𝐸

𝜕𝛼 
= ∑

1

𝛼+𝜆𝑥𝑖

𝑛
𝑖=1 − ∑ 𝑥𝑖 = 0

𝑛
𝑖=1                                                                         (15) 

𝜕ℓ𝐸𝑅
𝑀𝐿𝐸

𝜕𝜆
= ∑

𝑥𝑖

𝛼+𝜆𝑥𝑖

𝑛
𝑖=1 −

1

2
∑ 𝑥𝑖

2𝑛
𝑖=1 = 0                                                                     (16) 

          The maximum likelihood estimators denoted by �̂�𝑀𝐿𝐸(𝐸𝑅) and �̂�𝑀𝐿𝐸(𝐸𝑅) are the values 

of 𝛼  and 𝜆 that maximizes 𝐿(𝛼, 𝜆| 𝑥 )𝐸𝑅
𝑀𝐿𝐸 can be obtained by the solution of equations (15), 

(16). Note that there are no closed solutions of these equations; therefore, Newton – Raphson 

method is iterative technique can be applied to find the solution. 

          In Newton – Raphson method, the solution of the likelihood equation at iteration 

(ℎ + 1) is extract through the following iterative process, 



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(4)2021 
 

99 

 

[
�̂�

𝑀𝐿𝐸(𝐸𝑅)
(ℎ+1)

�̂�
𝑀𝐿𝐸(𝐸𝑅)
(ℎ+1)

] = [
�̂�

𝑀𝐿𝐸(𝐸𝑅)
(ℎ)

�̂�
𝑀𝐿𝐸(𝐸𝑅)
(ℎ)

] − 𝐽(ℎ)
−1

[
 
 
 
𝜕ℓ𝐸𝑅

𝑀𝐿𝐸

𝜕𝛼 
𝜕ℓ𝐸𝑅

𝑀𝐿𝐸

𝜕𝜆 ]
 
 
 

 𝛼=�̂�
𝑀𝐿𝐸(𝐸𝑅)
(ℎ)

 𝜆=�̂�
𝑀𝐿𝐸(𝐸𝑅)
(ℎ)

            ; ℎ = 0,1,2, … 

Where, 

𝐽(ℎ) =

[
 
 
 
 
𝜕2ℓ𝐸𝑅

𝑀𝐿𝐸

𝜕𝛼2
                 

𝜕2ℓ𝐸𝑅
𝑀𝐿𝐸

𝜕𝛼𝜕𝜆
𝜕2ℓ𝐸𝑅

𝑀𝐿𝐸

𝜕𝜆𝜕𝛼
                

𝜕2ℓ𝐸𝑅
𝑀𝐿𝐸

𝜕𝜆2
 ]

 
 
 
 

 
𝛼=�̂�

𝑀𝐿𝐸(𝐸𝑅)
(ℎ)

𝜆=�̂�
𝑀𝐿𝐸(𝐸𝑅)
(ℎ)

 

Where the first partial derivatives as in equations (15), (16) and the second partial 

derivatives are obtained as follows, 

𝜕2ℓ𝐸𝑅
𝑀𝐿𝐸

𝜕𝛼2
= − ∑

1

(𝛼+𝜆𝑥𝑖)
2

𝑛
𝑖=1                                                               (17) 

𝜕2ℓ𝐸𝑅
𝑀𝐿𝐸

𝜕𝜆2
= − ∑

𝑥𝑖
2

(𝛼+𝜆𝑥𝑖)
2

𝑛
𝑖=1                                                               (18) 

𝜕2ℓ𝐸𝑅
𝑀𝐿𝐸

𝜕𝛼𝜕𝜆
=  

𝜕2ℓ𝐸𝑅
𝑀𝐿𝐸

𝜕𝜆𝜕𝛼
= − ∑

𝑥𝑖

(𝛼+𝜆𝑥𝑖)
2

𝑛
𝑖=1                                                (19) 

          Now, based on an invariant property of the  𝑀𝐿𝐸 estimator, the survival function at 

mission time (t) of the 𝐸𝑅 distribution can be obtained by replacing 𝛼 and 𝜆  in equation (7), 

by their 𝑀𝐿𝐸 estimators as follows: 

�̂�(𝑡; 𝛼, 𝜆)𝐸𝑅
𝑀𝐿𝐸 = 𝑒

−(�̂�𝑀𝐿𝐸(𝐸𝑅)𝑡 + 
�̂�𝑀𝐿𝐸(𝐸𝑅)

2
 𝑡2)

                                      (20) 

 

3. Real-Life Data Applications 

3.1  Practical Application (1) 

     In this section, real data for lung cancer disease is analyzed, because of the importance of 

this disease, we have collected data related to mortality from this disease from the Iraqi 

Ministry of Health and Environment, Department of Medical City, Tumor Teaching 

Hospital, from 1 / 1 / 2015 to 1 / 1 / 2021. The data related to this disease were not taken 

during 2021 due to the spread of the Covid-19 epidemic, as patients do not stay in the 

hospital for more than a day or two for fear of contracting this epidemic. The sample size 

consists (100) observations. It was noted that all patients died during different periods and 

this means that the data or sample used is a complete data set.  

The following real data set represents the number of days of entering a patient with lung 

cancer to the hospital until death. The data set consist (100) observations: 



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(4)2021 
 

100 

 

[ 4  28  3  6  2  6  4  21  5  10  5  8  3  2  5  10  4  2  7  13  6  2  11  5  7  10  5  21  12  3  13  

12  2  10  4  3    22  2  7  20  18  4  10  5  16  21  4  26  12  15  3  5  25  3  15  20  13   6  22  

27  13  6  8  6  7  5  11  6  17  5  4  3  21  18  9  13  3  18  24  18  4  10  20  7  12  17  19  4  6  

15  15  4  6  6  12  11  3  10  2  29 ]. 

Using the maximum likelihood estimation in practical application to calculate the estimated 

value for the two parameters 𝛼 and 𝜆 as follows; 

          Number of Observation                                                                   100 

            Initial Value of 𝛼                                                                           0.01 
            Initial Value of 𝜆                                                                           0.01 
         Estimated Value of 𝛼                                                                      0.0412                            
         Estimated Value of 𝜆                                                                      0.0074 
 

The following graphic shows the frequency histogram table for lung cancer patients, the 

vertical axis represents the number of patients and the horizontal axis represents the number 

of days from the patient's admission to the hospital until his death 

 

Figure 9: Frequency Histogram Table for Lung Cancer Patients 

Now, we'll compute the expected frequency 𝐸𝑖 as follows; 

𝐸1 = 𝐹(3) × 100 

      = [1 − 𝑒
−[0.0412(3) + 

0.0074 (9)

2 ] × 100  

𝐸1 = 14.6 

𝐸2 = [𝐹(6) − 𝐹(3)] × 100 

      = [[1 − 𝑒
−[0.0412(6) + 

0.0074 (36)

2  ] − [1 − 𝑒
−[0.0412(3) + 

0.0074 (9)

2 ]] × 100  

𝐸2 = 17.1 

𝐸3 = [𝐹(9) − 𝐹(6)] × 100 

      = [[1 − 𝑒
−[0.0412(9) + 

0.0074 (81)

2  ] − [1 − 𝑒
−[0.0412(6) + 

0.0074 (36)

2 ]] × 100  

𝐸3 = 17.2 

N
o

. 
o

f 
p

a
ti

e
n

ts
 

D

D 

             Days 



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(4)2021 
 

101 

 

𝐸4 = [𝐹(12) − 𝐹(9)] × 100 

      = [[1 − 𝑒
−[0.0412(12) + 

0.0074 (144)

2  ] − [1 − 𝑒
−[0.0412(9) + 

0.0074 (81)

2 ]] × 100  

𝐸4 = 15.3 

𝐸5 = [𝐹(15) − 𝐹(12)] × 100 

      = [[1 − 𝑒
−[0.0412(15) + 

0.0074 (225)

2  ] − [1 − 𝑒
−[0.0412(12) + 

0.0074 (144)

2 ]] × 100  

𝐸5 = 12.4 

𝐸6 = [𝐹(18) − 𝐹(15)] × 100 

      = [[1 − 𝑒
−[0.0412(18) + 

0.0074 (324)

2  ] − [1 − 𝑒
−[0.0412(15) + 

0.0074 (225)

2 ]] × 100  

𝐸6 = 9.1 

𝐸7 = [𝐹(21) − 𝐹(18)] × 100 

      = [[1 − 𝑒
−[0.0412(21) + 

0.0074 (441)

2  ] − [1 − 𝑒
−[0.0412(18) + 

0.0074 (324)

2 ]] × 100  

𝐸7 = 6.1 

𝐸8 = [𝐹(24) − 𝐹(21)] × 100 

      = [[1 − 𝑒
−[0.0412(24) + 

0.0074 (576)

2  ] − [1 − 𝑒
−[0.0412(21) + 

0.0074 (441)

2 ]] × 100  

𝐸8 = 3.8 

𝐸9 = [𝐹(27) − 𝐹(24)] × 100 

      = [[1 − 𝑒
−[0.0412(27) + 

0.0074 (729)

2  ] − [1 − 𝑒
−[0.0412(24) + 

0.0074 (576)

2 ]] × 100  

𝐸9 = 2.2 

𝐸10 = [𝐹(30) − 𝐹(27)] × 100 

      = [[1 − 𝑒
−[0.0412(30) + 

0.0074 (900)

2  ] − [1 − 𝑒
−[0.0412(27) + 

0.0074 (729)

2 ]] × 100  

𝐸10 = 1.2 

Now, we’ll compute 

𝜒2 = ∑
(𝑂𝑖 − 𝐸𝑖)

2

𝐸𝑖

𝑘

𝑖=1

 



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(4)2021 
 

102 

 

𝜒2 =
(7 − 14.6)2

14.6
+  

(28 − 17.1)2

17.1
+

(17 − 17.2)2

17.2
+  

(11 − 15.3)2

15.3
+ 

(10 − 12.4)2

12.4
 

         + 
(7 − 9.1)2

9.1
+  

(8 − 6.1)2

6.1
+  

(6 − 3.8)2

3.8
+  

(3 − 2.2)2

2.2
+

(3 − 1.2)2

1.2
  

𝜒2 = 17.915 

It is discovered that the calculated value is (17.915); when comparing this value with 

tabulated value at the level of significance (0.01) and degrees of freedom (9) we find out that 

the calculated value is less than the tabulated value (21.67). That means accepting the null 

hypothesis 𝐻0 and the data is distributed according to Exponential Rayleigh 𝐸𝑅 distribution. 

Now, using estimate values for two parameters in Exponential Rayleigh distribution by 

𝑀𝐿𝐸  method to find numerical values for probability density function 𝑓(𝑡), cumulative 

distribution function �̂�(𝑡), survival function �̂�(𝑡) and hazard rate function ℎ̂(𝑡) as follows; 

Table 1 : Estimated values for functions 𝑓(𝑡),  �̂�(𝑡), �̂�(𝑡) and  ℎ̂(𝑡) 

t               𝑓𝑖 𝑓(𝑡) �̂�(𝑡) �̂�(𝑡) ℎ̂(𝑡) 
   2              7 0.0507 0.093 0.907         0.056 

   3              9 0.054 0.146 0.854 0.0634 

   4             10 0.056 0.201 0.799 0.0708 

   5              9 0.057 0.259 0.741 0.0782 

   6             10 0.0584 0.317 0.683 0.0856 

   7              5 0.0581 0.375 0.625         0.093 

   8              2 0.056 0.433 0.567 0.1004 

   9              1 0.055 0.489 0.511 0.1078 

  10             7 0.052 0.543 0.457 0.1152 

  11             3 0.049 0.594 0.406 0.1226 

  12             5 0.046 0.642 0.358 0.13 

  13             5 0.043 0.687 0.313 0.1374 

  15             4 0.035 0.766 0.234 0.1522 

  16             1 0.031 0.8 0.2 0.1596 

  17             2 0.028 0.83 0.17 0.167 

  18             4 0.024 0.857 0.143 0.1744 

  19             1 0.021 0.88 0.12 0.1818 

  20             3 0.018 0.901 0.099 0.1892 

  21             4 0.016 0.918 0.082 0.1966 

  22             2 0.013 0.933 0.067 0.204 

  24             1 0.0096 0.956 0.044 0.2188 

  25             1 0.0079 0.965 0.035 0.2262 

  26             1 0.0065 0.972 0.028 0.2336 

  27             1 0.0053 0.978 0.022 0.241 

  28             1 0.0042 0.983 0.017 0.2484 

  29             1 0.0033 0.987 0.013 0.2558 

 

Here, we will discuss the following important notes on the previous results table (1): 

1. The values of probability density function are increasing until 𝑡 = 6. Then the 

probability density function are decreasing when the failure times ( 7 ≤ 𝑡 ≤ 29 ),  so 

𝑡 = 6 is the mode of this function. Noting that the differences between all the values 

of probability density function  are very small and converged. 



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(4)2021 
 

103 

 

2. The cumulative distribution function values are increasing with the increase of failure 

times for the lung cancer patients in the hospital. This means there is a direct 

relationship between failure times and cumulative distribution function. 

3. The values of the probability of survival for patients are great at small failure times 

and vice versa. There is a reverse relationship between failure times and survival 

function. This means the values of the survival function �̂�(𝑡) are decreasing with the 

increasing of the failure times. So the patient who remains (𝑡 = 2) days has greatest 

probability of survival with (0.907), but the patient who remains (𝑡 = 29) days in a 

hospital has the smallest probability of survival with (0.013). 

4. The values of hazard function ℎ̂(𝑡) are increasing with the increasing of the failure 

times for the lung cancer patients in the hospital. This means there is a direct 

relationship between the failure times and hazard function. Noting that the 

probability of hazard for patients is small when small failure times and vice versa.  

The patient who remains (𝑡 = 2) days in a hospital has a smallest probability of 

hazard for death with (0.056), while the patient who remains (𝑡 = 29) days in a 

hospital has a greatest probability of hazard for death with (0.2558). 

3.2 Practical Application (2)  

     Stomach cancer is an abnormal growth of cells that begins in the stomach. Because of the 

importance of this disease, we have collected data related to mortality from this disease from 

the Iraqi Ministry of Health and Environment, Department of Medical City, Tumor Teaching 

Hospital, from  1 / 1 / 2015 to 1 / 1 / 2021.The data related to this disease were not taken 

during the year 2021 due to the spread of the Covid-19 epidemic, as patients do not stay in 

the hospital for more than a day or two for fear of contracting this epidemic. The sample size 

consists (81) observations. It was noted that all patients died during different periods and this 

means that the data or sample used is a complete data set.  

The following real data set represents the number of days from entering a patient with 

stomach cancer to the hospital until death. The data set consist (81) observations: 

[6, 11, 27, 5, 2, 16, 8, 25, 7, 26, 14, 8, 19, 5, 24, 22, 16, 20, 17, 11, 3, 15, 2, 6, 15, 29, 3, 24, 

15, 7, 3, 13, 19, 6, 4, 2, 11, 16, 27, 25, 15, 12, 7, 26, 9, 4, 2, 16, 22, 17, 6, 24, 23, 11, 15, 20, 

6, 4, 22, 10, 8, 4, 13, 9, 7, 2, 5, 26, 14, 19, 25, 22, 4, 9, 3, 8, 13, 27, 25, 16, 8]. 

Using the maximum likelihood estimation in practical application to calculate the estimated 

value for the two parameters 𝛼 and 𝜆 as follows; 

           Number of Observation                                                                      81 

              Initial Value of 𝛼                                                                            0.01 
              Initial Value of 𝜆                                                                             0.01 
           Estimated Value of 𝛼                                                                       0.0182 
           Estimated Value of 𝜆                                                                        0.0063 
 

The following graphic shows the frequency histogram table for stomach cancer patients, the 

vertical axis represents the number of patients and the horizontal axis represents the number 

of days from the patient's admission to the hospital until his death. 



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(4)2021 
 

104 

 

 
 

Figure 10: Frequency Histogram Table for Stomach Cancer Patients 

 

Now, we'll compute the expected frequency 𝐸𝑖 as follows; 

𝐸1 = 𝐹(3) × 81 

      = [1 − 𝑒
−[0.0182(3) + 

0.0063(9)

2 ] × 81  

𝐸1 = 6.48 

𝐸2 = [𝐹(6) − 𝐹(3)] × 81 

      = [[1 − 𝑒
−[0.0182(6) + 

0.0063 (36)

2  ] − [1 − 𝑒
−[0.0182(3) + 

0.0063 (9)

2 ]] × 81  

𝐸2 = 9.72 

𝐸3 = [𝐹(9) − 𝐹(6)] × 81 

      = [[1 − 𝑒
−[0.0182(9) + 

0.0063 (81)

2  ] − [1 − 𝑒
−[0.0182(6) + 

0.0063 (36)

2 ]] × 81  

𝐸3 = 11.583 

𝐸4 = [𝐹(12) − 𝐹(9)] × 81 

      = [[1 − 𝑒
−[0.0182(21) + 

0.0063 (144)

2  ] − [1 − 𝑒
−[0.0182(9) + 

0.0063 (81)

2 ]] × 81  

𝐸4 = 11.907 

𝐸5 = [𝐹(15) − 𝐹(12)] × 81 

      = [[1 − 𝑒
−[0.0182(15) + 

0.0063(225)

2  ] − [1 − 𝑒
−[0.0182(12) + 

0.0063(144)

2 ]] × 81  

𝐸5 = 11.016 

𝐸6 = [𝐹(18) − 𝐹(15)] × 81 

      = [[1 − 𝑒
−[0.0182(18) + 

0.0063(324)

2  ] − [1 − 𝑒
−[0.0182(15) + 

0.0063(225)

2 ]] × 81  

𝐸6 = 9.315 

𝐸7 = [𝐹(21) − 𝐹(18)] × 81 

      = [[1 − 𝑒
−[0.0182(21) + 

0.0063(441)

2  ] − [1 − 𝑒
−[0.0182(18) + 

0.0063(324)

2 ]] × 81  

𝐸7 = 7.209 

𝐸8 = [𝐹(24) − 𝐹(21)] × 81 

      = [[1 − 𝑒
−[0.0182(24) + 

0.0063(576)

2  ] − [1 − 𝑒
−[0.0182(21) + 

0.0063(441)

2 ]] × 81  

𝐸8 = 5.265 

𝐸9 = [𝐹(27) − 𝐹(24)] × 81 

      = [[1 − 𝑒
−[0.0182(27) + 

0.0063(729)

2  ] − [1 − 𝑒
−[0.0182(24) + 

0.0063(576)

2 ]] × 81  

N
o

. 
o

f 
P

a
ti

e
n

ts
 

            Days 



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(4)2021 
 

105 

 

𝐸9 = 3.564 

𝐸10 = [𝐹(30) − 𝐹(27)] × 81 

      = [[1 − 𝑒
−[0.0182(30) + 

0.0063(900)

2  ] − [1 − 𝑒
−[0.0182(27) + 

0.0063(729)

2 ]] × 81  

𝐸10 = 2.187 

Now, we’ll compute 

𝜒2 = ∑
(𝑂𝑖 − 𝐸𝑖)

2

𝐸𝑖

𝑘

𝑖=1

 

𝜒2 =
(5 − 6.48)2

6.48
+ 

(12 − 9.72)2

9.72
+

(14 − 11.583)2

11.583
+  

(8 − 11.907)2

11.907
+  

(6 − 11.016)2

11.016
 

     + 
(12 − 9.315)2

9.315
+  

(5 −7.209)2

7.209
+  

(5 − 5.265)2

5.265
+  

(10 − 3.564)2

3.564
+

(4 − 2.187)2

2.187
          

𝜒2 = 19.526 

It is discovered that the calculated value is (19.526), when comparing this value with the 

tabulated value at the level of significance (0.01) and degrees of freedom (9), we find out 

that the calculated value is less than the tabulated value (21.67). That means accepting the 

null hypothesis 𝐻0 and the data is distributed according to Exponential Rayleigh 

𝐸𝑅 distribution . 

Now, using estimate values for two parameters in Exponential Rayleigh distribution by 

𝑀𝐿𝐸  method to find numerical values for probability death density function 𝑓(𝑡), 

cumulative distribution function �̂�(𝑡), survival function �̂�(𝑡) and hazard rate function ℎ̂(𝑡) 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(4)2021 
 

106 

 

Table 2 : Estimated values for functions 𝑓(𝑡),  �̂�(𝑡), �̂�(𝑡) and  ℎ̂(𝑡) 

t           𝑓𝑖 𝑓(𝑡) �̂�(𝑡) �̂�(𝑡) ℎ̂(𝑡) 
    2          5 0.0293 0.0479 0.9521 0.0308 

    3          4  0.03414 0.0797 0.9203 0.0371 

    4          5 0.0383       0.116       0.884 0.0434 

    5          3 0.0419 0.1562 0.8438 0.0497 

    6          5 0.0448 0.1996 0.8004       0.056 

    7          4 0.0469 0.2456 0.7544 0.0623 

    8          5 0.0484 0.2934 0.7066 0.0686 

    9          3 0.0492 0.3423 0.6577 0.0749 

   10         1 0.0493 0.3917 0.6083 0.0812 

   11         4 0.0489 0.4409 0.5591 0.0875 

   12         1 0.0478 0.4894 0.5106 0.0938 

   13         3 0.0463 0.5366 0.4634 0.1001 

   14         2 0.0444 0.5819 0.4180 0.1064 

   15         5 0.0422 0.6254 0.3746 0.1127 

   16         5 0.0396 0.6664 0.3336       0.119 

   17         2 0.0370 0.7047 0.2953 0.1253 

   19         3 0.0312 0.7731 0.2269 0.1379 

   20         2 0.0284 0.8029 0.1971 0.1442 

   22         4 0.0228 0.8542 0.1458 0.1568 

   23         1 0.0202 0.8757 0.1243 0.1631 

   24         3 0.0178 0.8948 0.1052 0.1694 

   25         4 0.0155 0.9115 0.0885 0.1757 

   26         3 0.0134 0.9259 0.0740       0.182 

   27         3 0.0115 0.9385 0.0615 0.1883 

   29         1 0.0083 0.9583 0.0417 0.2009 

 

Here we will discussion the following important notes on the previous results table (2): 

1. The values of probability density function are increasing until 𝑡 = 10. Then the 

probability density function are decreasing when the failure times ( 11 ≤ 𝑡 ≤ 29 ),  so 

𝑡 = 10  is the mode of this function. Noting that the differences between all the values of 

probability density function are very small and converged. 

2. The cumulative distribution function values are increasing with the increase of failure 

times for the stomach cancer patients in the hospital. This means there is a direct 

relationship between failure times and cumulative distribution function. 

3. The values of the probability of survival for patients are great at small failure times and 

vice versa. There is a reverse relationship between failure times and survival function. 

This means the values of the survival function �̂�(𝑡) are decreasing with the increasing of 

the failure times. So the patient who remains (𝑡 = 2) days has greatest probability of 

survival with (0.9521), but the patient who remains (𝑡 = 29) days in a hospital has the 

smallest probability of survival with (0.0417). 

4. The values of hazard function ℎ̂(𝑡) are increasing with the increasing of the failure times 

for the stomach cancer patients in the hospital. This means there is a direct relationship 

between the failure times and hazard function. Noting that the probability of hazard for 

patients is short when small failure times and vice versa, The patient who remains (𝑡 =

2) days in a hospital has a smallest probability of hazard for death with (0.0308), while 



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(4)2021 
 

107 

 

the patient who remains (𝑡 = 29) days in a hospital has a greatest probability of hazard 

for death with (0.2009). 

 

4. Conclusion 

     Based on real data of lung cancer and stomach cancer the differences between all the 

values of probability density function 𝑓(𝑡) are very small and converged. The values of 

cumulative distribution function �̂�(𝑡) are increasing with the increasing of failure times for 

the patients in the hospital. The values of the probability of survival for patients was great at 

small failure times and vice versa. There is a reverse relationship  between failure times and 

survival function �̂�(𝑡). The values of hazard function ℎ̂(𝑡) are increasing with the increasing 

of the failure times for the patients in the hospital, that means there is a direct relationship 

between the failure times and hazard function.  

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