EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 1, No. 4, 2008, (30-40)
ISSN 1307-5543 – www.ejpam.com

Generalized residual entropy function and its applications

Mirza Abdul Khalique Baig1,∗, Javid Gani Dar1

1 P. G. Department of Statistics, University of Kashmir, Srinagar - 190006, India

Abstract. Shannon’s entropy plays an important role in the context of the information theorey. Since,
this entropy is not applicable to a system which has survived for some unit of time. So, the concept of
residual entropy was developed. In this paper, we study generalized information measure for residual
life time distributions and characterize some life time models based on this measure. Also, a new
classes of life time distributions are defined.
AMS subject classifications: 60E15, 62N05, 90B25, 94A17, 94A24
Key words: Varma’s entropy function, Life time distributions, Residual entropy.

1. Introduction

Let T be a continuous random variable with probability density function f (t), Varma’s
entropy of order α and type β is defined by

Hν(α,β) =
1

β −α
log

∫

f α+β−1(t)d t f or β − 1< α < β , β ≥ 1. (1.1)

and in discrete case

Hν(α,β) =
1

β −α
log

 

n
∑

k=1

Pα+β−1
k

!

f or β − 1< α < β , β ≥ 1. (1.2)

Also

lim
α→1,β=1

Hν(α,β) =−
∫

f (t) log f (t)d t (1.3)

and in discrete case

lim
α→1,β=1

Hν(α,β) =−
n
∑

k=1

Pk log Pk (1.4)

which is Shannon’s entropy in both the cases.
Varma’s entropy plays a vital role as a measure of complexity and uncertainty in different

areas such as physics, electronics and engineering to describe many chaotic systems.

∗Corresponding author. Email addresses: baigmak@yahoo.co.in (M. A. K. Baig)
jvdevi@gmail.com (J.Devi)

http://www.ejpam.com 30 c© 2008 EJPAM All rights reserved.



M. A. K. Baig and J. D. Gar / Eur. J. Pure Appl. Math, 1 (2008), (30-40) 31

As argued by Ebrahimi[4], if a unit is known to have survived up to an age t, then H(t) is
no longer useful in measuring the uncertainty about the remaining life time of the unit. The
idea is that a unit with great uncertainty is less reliable than a unit with low uncertainty. Ac-
cordingly, he introduced a measure of uncertainty known as residual entropy for the residual
life time distribution. The residual entropy of continuous random variable T is defined as

H(T, t) =−
∫ ∞

t

f (x)
R(t)

log
f (x)
R(t)

d x (1.5)

and in case of discrete random variable

H(t j) =−
n
∑

k= j

P(tk)
R(t j)

log
P(tk)
R(t j)

(1.6)

where R(t) is the reliability function of the random variable T.

2. Generalized Residual Entropy Function:

Let T be the non negative random variable representing component failure time with
failure distribution F(t) = P(T ≤ t) and survival function R(t) = 1− F(t) with R(0) = 1. We
define Varma’s entropy for residual life as

Hν(α,β , t) =
1

β −α
log





∫∞
t

f α+β−1(x)

Rα+β−1(t)
d x



 , β − 1< α < β , β ≥ 1. (2.1)

or

(β −α)Hν(α,β , t) = log

�
∫ ∞

t

f α+β−1(x)d x

�

− (α+ β − 1) log R(t), β − 1< α < β ,β ≥ 1.

(2.2)
for β = 1, α→ 1, (7) reduces to (5).
We now show that Hν(α,β , t) uniquely determines the R(t).

THEOREM 2.1: Let T be the non negative random variable having continuous density
function f and distribution function F with survival function R(t). Assume Hν(α,β , t) <
∞, t ≥ 0,β − 1 < α < β , β ≥ 1 and increasing in t, then Hν(α,β , t) uniquely determines
R(t).
Proof: Differentiating (8) with respect to t, we have

(β −α)H
′

ν(α,β , t) = (α+ β − 1)h(t)−
f α+β−1(t)

∫∞
t

f α+β−1(x)d x
(2.3)

where h(t) = f (t)
R(t) is the failure rate function.

From (8) and (9), we have

hα+β−1(t) = (α+ β − 1)h(t)exp
�

(β −α)Hν(α,β , t)
�

(2.4)

− (β −α)H
′

ν(α,β , t)exp((β −α)Hν(α,β , t)).



M. A. K. Baig and J. D. Gar / Eur. J. Pure Appl. Math, 1 (2008), (30-40) 32

Hence for fixed t > 0, h(t) is a solution of

g(x) = (x)α+β−1− (α+ β − 1)x exp
�

(β −α)Hν(α,β , t)
�

(2.5)

+ (β −α)H
′

ν(α,β , t)exp
�

(β −α)Hν(α,β , t)
�

= 0.

Differentiating both sides with respect to x , we have

g
′
(x) = (α+ β − 1)(x)α+β−2− (α+ β − 1)exp

�

(β −α)Hν(α,β , t)
�

. (2.6)

For extreme value of g(x), we have
g
′
(x) = 0, which gives

x = exp
�

β −α
α+ β − 2

Hν(α,β , t)
�

= x t

Also
g
′′
(x) = (α+ β − 1)(α+ β − 2)xα+β−3

Case I: Let α+ β > 2, then g
′′
(x t) > 0. Thus g(x) attains minimum at x t . Also, g(0 > 0

and g(∞) =∞. Further, g(x) decreases for 0 < x < x t and hence increases for x > x t . So,
x = h(t) is the unique solution to g(x) = 0.
Case II: Let α+β < 2, then g

′′
(x t)< 0. Thus g(x) attains maximum at x t . Also, g(0> 0 and

g(∞) = −∞. Further, it can be easily seen that g(x) decreases for x > x t and increases for
0< x < x t . So, x = h(t) is the unique solution to g(x) = 0.
Remark: For β = 1, x t = exp(−Hν(α, t)), which is given by Baig and Dar[2].
Corollary 2.1: If Hν(α,β , t) is decreasing in t, then (11) has a unique solution if g(x t) = 0.
i.e, Hν(α,β , t) = (2−α−β

β−α ) log(b− t) which is the Varma’s residual entropy of order α and type
β of the uniform distribution over (a,b). Thus the uniform distribution can be characterized
by decreasing Varma’s residual entropy Hν(α,β , t) = (2−α−β

β−α ) log(b− t).

Proof: Hν(α,β , t) = (2−α−β
β−α ) log(b− t) is the Varma’s residual entropy of the uniform distri-

bution. By putting it in (11), we have g(x t) = 0. Hence Hν(α,β , t) = (2−α−β
β−α ) log(b − t) is

the unique solution to g(x t) = 0, which proves the theorem.
Remark: For β = 1, Hν(α, t) = log(b− t), which is given by Baig and Dar [2].
Corollary 2.2: Let T be the random variable having Varma’s entropy of order α and type β
with α+ β > 2, be of the form

Hν(α,β , t) =
1

β −α
log(k)−

2−α− β
β −α

log h(t) (2.7)

where h(t) is the failure rate function of T , then T has
I. Exponential distribution iff k = 1

α+β−1

II. Pareto distribution iff k < 1
α+β−1



M. A. K. Baig and J. D. Gar / Eur. J. Pure Appl. Math, 1 (2008), (30-40) 33

III. Finite range distribution iff k > 1
α+β−1

Proof: (I) Let T has exponential distribution with probability distribution function

f (t) =
1

θ
exp
�

−
t

θ

�

, t > 0,θ > 0

The reliability function is given by

R(t) = exp
�

−
t

θ

�

The failure rate function is

h(t) =
1

θ

Therefore

Hν(α,β , t) =
1

β −α
log





∫∞
t

f α+β−1(x)

Rα+β−1(t)
d x



 , β − 1< α < β , β ≥ 1

or

Hν(α,β , t) =
1

β −α
log(k)−

2−α− β
β −α

log h(t)

where k = 1
α+β−1

, h(t) = 1
θ

Thus (13) holds.
Conversely, suppose k = 1

α+β−1

1

β −α
log(k)−

2−α− β
β −α

log h(t) =
1

β −α
log





∫∞
t

f α+β−1(x)

Rα+β−1(t)
d x





or

∫ ∞

t

f α+β−1(x)d x = Rα+β−1(t)exp
�

log(k)− (2−α− β)log h(t)
�

Differentiating both sides with respect to t, we have

h2(t)

h′(t)
=

k(2−α− β)
1− k(α+ β − 1)



M. A. K. Baig and J. D. Gar / Eur. J. Pure Appl. Math, 1 (2008), (30-40) 34

or

h−2(t)h
′
(t) =

1− k(α+ β − 1)
k(2−α− β)

or

h(t) =
�

1− k(α+ β − 1)
k(α+ β − 2)

t +
1

h(0)

�−1

= (at + b)−1 (2.8)

where a = 1−k(α+β−1)
k(α+β−2) and b = 1

h(0) .

Now k = 1
α+β−1

, therefore a = 0.
Clearly (14) is the failur rate function of the exponential distribution.
(II) The density function of the Pareto distribution is given by

f (t) =
(b)

1
a

(at + b)1+
1
a

, t ≥ 0, a > 0, b > 0

The reliability function is given by

R(t) =
(b)

1
a

(at + b)
1
a

, t ≥ 0, a > 0, b > 0

The failure rate is given by
h(t) = (at + b)−1 (2.9)

and

Hν(α,β , t) =
1

β −α
log(k)−

2−α− β
β −α

log h(t)

where k = 1
(α+β−1)+a(α+β−2) and h(t) = (at + b)−1. Since α+ β > 2, therefore k < 1

α+β−1
Thus (13) holds.
Conversly, suppose k < 1

α+β−1
, proceeding as in (I), (14) gives

h(t) =
�

1− k(α+ β − 1)
k(α+ β − 2)

t +
1

h(0)

�−1

= (at + b)−1 (2.10)

where a =
�

1−k(α+β−1)
k(α+β−2)

�

and b = 1
h(0) .

Since k < 1
α+β−1

and α+ β > 2, therefore a > 0.
Clearly (16) is the failure rate function of the Pareto distribution given in (15).
(III) The density function of the finite range distribution is given by

f (t) =
β1

ν

�

1−
t

ν

�β1−1
,β1 > 1, 0≤ t ≤ ν <∞



M. A. K. Baig and J. D. Gar / Eur. J. Pure Appl. Math, 1 (2008), (30-40) 35

The reliability function is given by

f (t) =
�

1−
t

ν

�β1

, β1 > 1,0≤ t ≤ ν <∞

The failure rate function is given by

h(t) =
�

β1

ν

�

�

1−
t

ν

�−1
(2.11)

and

Hν(α,β , t) =
1

β −α
log(k)−

2−α− β
β −α

log h(t)

where k = β1

(α+β−1)(β1−1)+1
and h(t) =

�

β1

ν

�

�

1− t
ν

�−1
.

Since α+ β > 2, therefore k > 1
α+β−1

.
Thus (13) holds.
Conversely, suppose k > 1

α+β−1
. Proceeding as in (I), (14) gives

h(t) = h(0)
�

1−
k(α+ β − 1)− 1

k(α+ β − 2)
h(0)t

�−1

(2.12)

which is the failure rate function of the distribution given in (17), iff k > 1
α+β−1

.
Remark: For β = 1, (13), (14), (16), (18) reduces to

Hν(α, t) =
1

1−α
log(k)− log h(t), h(t) =

�

(1− kα)t
k(α− 1)

+
1

h(0)

�−1

,

h(t) =
�

(1− kα)t
k(α− 1)

+
1

h(0)

�−1

, h(t) = h(0)
�

1−
(kα− 1)h(0)t

k(α− 1)

�−1

respectively, which is given by Baig and Dar [2].

3. New Class Of Life Time Distribution:

The survival function has increasing(decreasing) Varma’s entropy for residual life of order
α and type β , IVERL(α,β)(DVERL(α,β)) if Hν(α,β , t) is increasing(decreasing) in t, t > 0.
This implies that R has IVERL(α,β)(DVERL(α,β)) if

H
′

ν(α,β , t) ≥ 0

≤ 0



M. A. K. Baig and J. D. Gar / Eur. J. Pure Appl. Math, 1 (2008), (30-40) 36

Theorem 3.1: If a distribution is IVERL(α,β) as well as DVERL(α,β) for some constant, then
it must be exponential.
Proof: Since the random variable T is both IVERL(α,β) and DVERL(α,β), therfore

Hν(α,β , t) = constant

1

β −α
log





∫∞
t

f α+β−1(x)

Rα+β−1(t)
d x



= k

or

∫ ∞

t

f α+β−1(x)d x = Rα+β−1(t)exp(k(β −α))

Differentiating both sides with respect to t, we get

f (t)
h(t)

= constant

or

h(t) = constant

which means that the distribution is exponential.
The next theorem gives upper(lower)bounds to the failure rate function.
Theorem 3.2: If T is IVERL(α,β)(DVERL(α,β)), then

(I) (h(t)≤ (≥)(α+ β − 1)
1

α+β−2 exp
�

− α−β
α+β−2

Hν(α,β , t)
�

if α+ β > 2.

(II) h(t)≥ (≤)(α+ β − 1)
1

α+β−2 exp
�

− α−β
α+β−2

Hν(α,β , t)
�

if α+ β < 2.
Proof : If T is IVERL(α,β), then

H
′

ν(α,β , t)≥ 0

which gives

hα+β−2(t)≤ (α+ β − 1)exp
�

(β −α)Hν(α,β , t)
�

.



M. A. K. Baig and J. D. Gar / Eur. J. Pure Appl. Math, 1 (2008), (30-40) 37

Similarly, if T is DVERL(α,β), then

hα+β−2(t)≥ (α+ β − 1)exp
�

(β −α)Hν(α,β , t)
�

.

Case I: If α+ β > 2 and T is IVERL(α,β)(DVERL(α,β)), then

h(t)≤ (≥)(α+ β − 1)
1

α+β−2 exp
�

−
α− β

α+ β − 2
Hν(α,β , t)

�

(3.1)

Case II: If α+ β < 2 and T is IVERL(α,β)(DVERL(α,β)), then

h(t)≥ (≤)(α+ β − 1)
1

α+β−2 exp
�

−
α− β

α+ β − 2
Hν(α,β , t)

�

(3.2)

Remark: For β = 1, (19) reduces to

h(t)≤ (≥)(α)
1
α−1 exp

�

−Hν(α, t)
�

, which is given by Baig and Dar[2].
Remark : For β = 1, α→ 1 (19) reduce to
h(t)≤ (≥)exp(−H(T, t)), which is given by Ebrahimi [4].

4. Applications:

Let T be a discrete random variable taking values t1, t2, · · · , tn with respective probabilities
p1, p2, · · · , pn. The discrete residual entropy is defined as

H(P, j) =−
n
∑

k= j

pk

R( j)
log
�

pk

R( j)

�

(4.1)

The Verma’s residual entropy for discrete case is defined as

Hν(α,β , j) =
1

β −α
log







n
∑

k= j

pα+β−1
k

Rα+β−1( j)






(4.2)

for β = 1,α→ 1, (22) reduces to (21).
Theorem 4.1: If T has a discrete distribution F(t) with support (t j : t j < t j+1) and an increas-
ing Varma’s entropy Hν(α,β , t), then Hν(α,β , t) uniquely determines F(t).
Proof: We have

Hν(α,β , j) =
1

β −α
log







n
∑

k= j

pα+β−1
k

Rα+β−1( j)







or
n
∑

k= j

Pα+β−1
k = Rα+β−1( j)exp((β −α)Hν(α,β , j)) (4.3)



M. A. K. Baig and J. D. Gar / Eur. J. Pure Appl. Math, 1 (2008), (30-40) 38

For j+ 1, we have

n
∑

k= j+1

Pα+β−1
k = Rα+β−1( j+ 1)exp((β −α)Hν(α,β , j+ 1)) (4.4)

Subtracting (24) from (23), we have

Pα+β−1
j = Rα+β−1( j)exp((β −α)Hν(α,β , j))− Rα+β−1( j+ 1)exp((β −α)Hν(α,β , j+ 1))

Using Pj = R( j)− R( j+ 1), we get

(R( j)−R( j+1))α+β−1 = Rα+β−1( j)exp((β−α)Hν(α,β , j))−Rα+β−1( j+1)exp((β−α)Hν(α,β , j+1))

or
exp((β −α)Hν(α,β , j)) = (1− h j)

α+β−1+ hα+β−1
j exp((β −α)Hν(α,β , j+ 1))

where h j =
R( j+1)

R( j) ∈ (0, 1), which is the solution of the following equation

g(x) = (1− x)α+β−1+ xα+β−1 exp((β −α)Hν(α,β , j+ 1)) (4.5)

− exp((β −α)Hν(α,β , j)) = 0

Differentiating both sides with respect to x , we have

g
′
(x) = −(α+ β − 1)(1− x)α+β−2 (4.6)

+ (α+ β − 1)xα+β−2 exp((β −α)Hν(α,β , j+ 1))

Note that g
′
(x) = 0, gives

x =
�

1+ exp
�

β −α
α+ β − 2

Hν(α,β , j+ 1)
��−1

= x j

Further, from (25) we have g(0)≤ 0 and g(1)≥ 0.
Case I : Let α+ β > 2, then
g
′
(x)> 0 if x < x j

g
′
(x) = 0 if x = x j

g
′
(x)< 0 if x > x j

which implies that g(x) = 0 has a unique solution h j ∈ (0,1).
Case II : Let α+ β < 2, then
g
′
(x)> 0 if x > x j

g
′
(x) = 0 if x = x j

g
′
(x)< 0 if x < x j

which again shows that g(x) = 0 has a unique solution h j ∈ (0,1). Combining both the cases,
we conclude that the unique solution to g(x) = 0 is given by x = h j .
Thus Hν(α,β , j) uniquely determines F(t).



REFERENCES 39

Remark: Forβ = 1, x j =
�

1+ exp(−Hν(α, j+ 1)
�−1 which is given by Baig and Dar [2]

Theorem 4.2: The discrete uniform distribution is characterized by Varam’s residual entropy

Hν(α,β , j) =
�

2−α− β
β −α

log(n− j+ 1)
�

, j = 1, 2, ..., n

Proof: By putting Hν(α,β , j) =
�

2−α−β
β−α log(n− j+ 1)

�

, j = 1,2, ..., n in (25), we have

g(x j) = 0. Hence Hν(α,β , j) =
�

2−α−β
β−α log(n− j+ 1)

�

, j = 1, 2, ..., n is the unique solu-
tion to g(x j) = 0. Hence the theorem follows.
Remark: For β = 1, Hν(α, j) = log(n− j + 1), j = 1, 2, · · · , n which is given by Baig and Dar
[2].

5. Conclusion:

We introduce and studied the concept of Varma’s entropy for the life time distributions
that generalizes the entropy measure given by Ebrahimi[4]. The exponential, the Pareto
and the finite Range distributions which are commonly used in the reliability modeling have
been characterized in terms of the Varma’s entropy. The proposed residual entropy function
uniquely determines the distribution function and thus the reliability function. Also, we char-
acterize the discrete uniform distribution in terms of discrete generalized entropy.

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