Characterization of Negative Exponential Distribution through Expectation ()
1. Introduction
Knowing characterizing property may provide unexpectedly accurate information about distributions and one can recognize a class of distributions before any statistical inference is made. This feature of characterization of probability distributions is peculiar to characterizing property and attracted attention of both theoretician and applied workers but there is no general theory of it.
Various approaches were used for characterization of negative exponential distribution. Among many other people, Fisz [1], Tanis [2], Rogers [3] and Fergusion [4] used properties of identical distributions, absolute continuity, constant regression of adjacent order statistics, linear regression of adjacent order statistics of random variables and characterized negative exponential distribution. Using independent and non-degenerate random variables Fergusion ([5,6]) and Crawford [7] characterized negative exponential distribution. Linear regression of two adjacent record values used by Nagaraja ([8,9],) were different from two conditional expectations, conditioned on a non-adjacent order statistics used by Khan [10] to characterize negative exponential distribution.
In this research note section 2 is devoted for characterization based on identity of distribution and equality of expectation function randomly variable for a negative exponential distribution with probability density function (pdf).
(1.1)
where 
are known as constants, 
is positive absolute continuous function and 
is everywhere differentiable function. Since derivative of 
is positive, the range is truncated by 
from left
.
2. Characterization
Theorem 2.1 Let 
be a random variable with distribution function
. Assume that
is continuous on the interval
, where
. Let 
and 
be two distinct differentiable and intregrable functions of X on the interval 
where 
and moreover 
be non constant. Then
(2.1)
is the necessary and sufficient condition for pdf 
of 
to be 
defined in (1.1).
Proof Given 
defined in (1.1), for necessity of (2.1) if 
is such that 
where 
is differentiable function then
(2.2)
Differentiating with respect to 
on both sides of (2.2), replacing 
for 
and simplifying one gets
(2.3)
which establishes necessity of (2.1). Conversely given (2.1), let 
be such that
(2.4)
which can be rewritten as
(2.5)
which reduces to
(2.6)
Hence
. (2.7)
Since 
is increasing integrable and differentiable function on the interval 
with 
the following identity holds
. (2.8)
Differentiating 
with respect to
and simplifying (2.8) after taking 
as one factor, (2.8) reduces to
, (2.9)
where 
is a function of 
only derived in (2.3) and 
is a function of 
and 
only derived in (2.7).
Since 
be increasing integrable and differentiable function on the interval 
where 
and since 
is positive intregrable function on the interval 
where 
with 
and integrating (2.7) over the interval 
on both sides, one gets (2.7) as
(2.10)
and
.
Substituting 
in 
derived in (2.10), 
reduces to 
defined in (1.1) which establishes sufficiency of (2.1).
Remark 2.1 Using 
derived in (2.3), the 
given in (1.1) can be determined by
(2.11)
and pdf is given by
(2.12)
where 
is decreasing function for 
with 
such that it satisfies
. (2.13)
Illustrative Example: Using method described in the remark characterization of negative exponential distribution through survival function 
is illustrated.
3. Conclusion
To characterize pdf defined in (1.1) one needs any arbitrary non-constant function of 
which should only be differentiable and integrable.
NOTES