Page 21 - Contributed Paper Session (CPS) - Volume 5

CPS649 A-Hadi N. Ahmed et al.
Corollary 1: If the random variable ~() then the mode of is located
at , where is a positive integer satisfies 1−3 ≤ ≤ 1−2 . This implies

that ( + 1) ≥ () ∀ ≤ ( + 2) ≤ ( + 1) ∀ ≥ .

Definition 4: A discrete life distribution = { = ( = )}, ∈ ,

where is the set of all non-negative integers. With = ( ≤ ), we

define the discrete reversed failure rate (DRFR) as follows

∗
= , ∈ .

Definition 5: (Al-Zahrani and Al-Sobhi, 2015): A discrete life distribution =
{ = ( = )}, ∈ , where is the set of all non-negative integers is

said to be discrete increasing (decreasing) reversed failure rate DIRFR (DDRFR)
if , ∈ is increasing (decreasing).
∗

Proposition 1: Let the sequence defined by the NDL distribution, then NDL

distribution has the DIRFR property.

∗
Proof. It is easy to prove that is increasing in . The reversed hrf of is

(; ) (1 + + )(1 − )
∗
(; ) = = .
1 − (; ) [(1 + ) − (1 + + )(1 − ) ]

Remark 1: The following is a simple recursion formula for ( + 1) in terms
of () of the NDL for = 0,1,2, …, where
3 +
2
( + 1) = (1 − )(), ℎ (0) = 2 /(1 + ).
2 +
Remark 2:
(i) (0) = (0).
(ii) () is an increasing function in and .
(iii) () ≥ (0) ∀ ∈ hence the NDL distribution has the new
better than used in failure rate (NBUFR) property (see, Abouammoh
and Ahmed (1988)).

3.2 Stochastic interpretations of the parameter theta
Stochastic orders are important measures to judge comparative behaviors of
random variables. Shaked and Shanthikumar (2007) showed that many
stochastic orders exist and have various applications.
The following chains of implication (see, e.g., Shaked and Shantihkumar, 2007)
hold.
≤ ℎ
≤ ⟹ ⇓ ⟹ ≤ and ≤ ⇒ ≤ ℎ

≤

Theorem 2: Let ~( ) and ~( ). Then, ≤ for all > .
2
1
2
1

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