MISHRA DISTRIBUTION

International Journal of Mathematics and Statistics Invention (IJMSI)
E-ISSN: 2321 – 4767 P-ISSN: 2321 - 4759
www.ijmsi.org Volume 3 Issue 8 || December. 2015 || PP-14-17
www.ijmsi.org 14 | Page
MISHRA DISTRIBUTION
Binod Kumar Sah
Associate Professor in Statistics,Department of Statistics, R.R.M. Campus, Janakpur, Tribhuvan University, Nepal
Email:sah.binod01@gmail.com
Abstract: We name this distribution as “Mishra distribution” in honor of Prof. A.Mishra, Department of
Statistics, Patna University, Patna. It is a new member of a family of exponential probability distributions. This
continuous probability distribution has a single parameter. Its probability density function has been obtained.
Its different characteristics such as moments about origin, Co-efficient of Variation, moment generating
function and probability distribution have been obtained. .Estimation of Parameter of this distribution has been
discussed by the method of moments as well as maximum likelihood method. The distribution has been fitted to
some data-sets to test its goodness of fit and it has been found that this distribution gives better fit than the
Exponential distribution and in some cases it gives better fit than the Lindley distribution. .
Key-words: Exponential distribution, Lindley distribution, moments, estimation of parameters, goodness of
fit.
I. INTRODUCTION
The exponential distribution occupies a central place among the continuous probability distributions
and plays an important role in statistical theory with many interesting properties and having a wide range of
applications in various fields. Its probability density function (pdf) is given by
0,0;);( 



xexf
x
… … (1.1)
One of the interesting properties of this distribution is its ‘memory loss’ which is expressed as
     = .P X s t P X s P X t    … … (1.2)
Its rth moment about origin is given by
 
...,3,2,1,
1/


 r
r
rr

 … … (1.3)
Lindley (1958) introduced a one-parameter distribution, known as Lindley distribution, given by its
probability density function
0,0;)1(
1
);(
2









xexxf
x
… (1.4)
The first four moments about origin of the Lindley distribution have been obtained as
)1(
)2(2/
1





 ,
)1(
)3(2
2
/
2





 ,
)1(
)4(6
3
/
3





 ,
)1(
)5(24
3
/
4





 (1.5)
Ghitany et al (2008) have discussed various properties of this distribution and showed that in many
ways (1.4) provides a better model for some applications than the exponential distribution. Mazucheli and
Achcar (2011), Ghitany et al (2009, 2011) and Bakouchi et al (2012) are some among others who discussed its
various applications. Zakerzadah and Dolati (2009), Shanker and Mishra (2013a,2013b) obtained generalized
Lindley distributions and discussed their various properties and applications. Sankaran (1970) obtained a
Lindley mixture of Poisson distribution.Sah B.K. (2015a) obtained a two-parameter Quasi-Lindley distribution
and discussed their various properties.
II. MISHRA DISTRIBUTION
A one-parameter Mishra distribution (MD) with parameters  is defined by its probability density
function (pdf)
0,0;
)2(
)1(
);( 2
23





x
exx
xf
x





… (2.1)

MISHRA DISTRIBUTION
III. MOMENTS
The rth
moment about origin of the Mishra distribution has been obtained as






0
2
2
3
/
)1(
)2(
dxexxx
xr
r



 … … … (3.1)
=
)2(
)}2)(1()1({!
2
2





rrrr
r
.. … … (3.2)
Putting the value r=1, 2, 3 and 4 in the expression (3.2), the first four moments about origin of the MD are
obtained as
)2(
)62(!1
2
2
/
1







)2(
)123(!2
2
2
2
/
2







)2(
)204(!3
2
2
3
/
3







)2(
)305(!4
2
2
4
/
4






 … (3.3)
The variance is given by
/ /2
2 2 1
   
=
100
)6(
)62()2)(123(2
2
2222





… … (3.4)
Co – efficient of Variation (C.V.)
C.V. = 100
)6(
)62()2)(123(2
2
2222





… … (3.5)
Moment Generating Function ( )x
M t
Moment generating function of Mishra distribution can be obtained as
 ( )x
M t
0
( )
tx
e f x d x

 









 3
2
2
3
)(
2)()(
)2( t
tt




… … (3.6)
Distribution Function
Distribution function of the Mishra distribution is obtained as
dxexxF
x
x
x


 


  )1(
)2(
2
0
2
3
=
)2(
)2(
1 2









x
x exx
e … … (3.7)

MISHRA DISTRIBUTION
IV. ESTIMATION OF PARAMETER
Here, we have discussed two methods (a) the method of moments, and (b) the method of maximum
likelihood to estimate parameter of the Mishra distribution.
(a) The method of moments
Parameter of the Mishra distribution can be obtained by using the first moment about origin.
)2(
)62(!1
2
2
/
1







622,
2/
1
2/
1
3/
1
 Or
06)1(2)1(,
/
1
2/
1
3/
1
 Or … … (4.1)
The expression (4.1) is the Polynomial in third degree equation. Replacing the corresponding population
moment by sample moment and solving the expression (4.1) by using Regula-Falsi method, we get an estimate
of .
(b) The method of maximum likelihood
Let ( 1 2
, , , n
x x x ) be a random sample of size n from a single parameter Mishra distribution and let
x
f be the observed frequency in the sample corresponding to X x  1, 2,...,x k such that
1
k
x
x
f n

 ,
where k is the largest observed value having non-zero frequency. The likelihood function, L of the Mishra
distribution (2.1) is given by














k
x
xnf
n
exxL x
1
2
2
3
)1(
2



... (4.2)
and so the log likelihood function is obtained as



k
x
x
xnxxfnnL
1
22
)1()2log(log3log  (4.3)
The log likelihood equation is thus obtained as
0
)2(
)12(3log
2






xn
nnL



… … … (4.4)
Or, x
)2(
)62(!1
2
2





Or, 06)1(2)1(
23
  xxx … … (4.5)
Solving the expression (4.5) by using Regula-Falsi method, we get an estimate of  .
V. GOODNESS OF FIT
The Mishra distribution has been fitted to a number of data- sets to which earlier the exponential
distribution and the Lindley distribution have been fitted by others and to almost all these data-sets the Mishra
distribution provides closer fits than the Exponential distribution. This distribution gives better fit to the second data
set than Lindley distribution.
The fittings of the Mishra distribution to two such data-sets have been presented in the following tables. The data
sets given in tables-I and II are the data sets reported by Ghitany et al (2008) and Bzerkedal (1960) respectively.
The expected frequencies according to the exponential distribution and the Lindley distribution have also been
given for ready comparison with those obtained by the Mishra distribution. The estimate of the parameter has been
obtained by the method of maximum likelihood.

MISHRA DISTRIBUTION
VI. CONCLUSION
In this paper, we propose, a single parameter continuous distribution, Mishra distribution (MD).
Several properties such as moments, moment generating function, distribution function have been obtained .The
methods of estimation of parameter have been discussed. Finally, the proposed distribution has been fitted to a
number of data-sets to test its goodness of fit and it has been observed that the MD gives better fit to all the data-
sets than the Exponential distribution. In some cases, it gives better fit than the Lindley distribution.
REFERENCES
[1] Bakouch,H.S., Al-Maharani, B.M., Al-Shomrani, A.A., Marchi, V.A.A., Louzada, F. (2012): Anextended Lindley distribution,
Journal of the Korean Statistical Society, Vol 41 (1), 75- 85.
[2] Bzerkedal,T. (1960): Acquisition of resistance in guinea pigs infected with different doses of virulent tubercle bacilli, American
Journal of Epidemiol, Vol. 72 (1), 130 – 148.
[3] Ghitany,M. E., Atieh, B., Nadarajah, S. (2008): Lindley distributionanditsApplications,Mathematics and Computers in
Simulation, Vol.78 (4), 493 – 506.
[4] Ghitany, M. E. Al-qallaf, F., Al-Mutairi, D.K., Hussain, H.A. (2011): A two parameter weighted Lindley distribution and its
applications to survival data, Mathematics and Computers in Simulation, Vol. 81 (6), 1190- 1201.
[5] Lindley, D.V. (1958): Fiducial distributions and Bayes’ theorem, Journal of the Royal Statistical
Society, Series B, 20, 102- 107.
[6] Mazucheli,J. and Achcar,J.A. (2011): The Lindley distribution applied to competing risks lifetime data, Computer Methods
and Programs in Biomedicine, Vol. 104 (2), 188 – 192.
[7] Sankaran, M. (1970): The discrete Poisson-Lindley distribution, Biometrics, 26, 145 – 149.
[8] Sah, B.K. (2015a): A two-parameter Quasi-Lindley distribution, Ideal Science Review, Vol.7 (1), pp. 16-18.
[9] Shanker, R. and Mishra, A. (2013a): A quasi Lindley distribution, African Journal of Mathematics andComputer Science
Research,Vol.6 (4), 64-71.
[10] Shanker, R and Mishra, A. (2013b): A two-parameter Lindley distribution, Statistics in Transition,(New Series),Vol.14 (1),45- 56
[11] Shaked,M. and Shanthikumar,J.G.(1994): Stochastic Orders and Their Applications, AcademicPress, New York.
[12] Zakerzadah, H. and Dolati, A. (2009): Generalized Lindley distribution, Journal of Mathematical Extension, Vol. 3(2), 13 – 25.

MISHRA DISTRIBUTION

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