Page 19 - Contributed Paper Session (CPS) - Volume 5

CPS649 A-Hadi N. Ahmed et al.
This new distribution can be considered as an alternative to the negative
binomial. Poisson-inverse Gaussian. hyper-Poisson and generalized Poisson
distributions.

Definition 1
Let be a non-negative discrete random variable obtained as a finite
mixture of geometric () and negative binomial (2, ) with mixing
probabilities and respectively. The new TNDL distribution is specified
+ +
by the pmf
2

(; , ) = (1 − ) [1 + (1 + )], = 0,1,2, … , ∈ (0,1).
+
We note that the TNDL distribution includes the following discrete
distributions as particular cases:
(i) The geometric distribution when = 0.
(ii) The two-parameter discrete Lindley distribution of Hussain et al.
(2016), when = 1 − and = 0.5.
(iii) The two-parameter discrete Lindley distribution of Bakouch et al.
(2014), when = 1 − .

The corresponding survival function (sf) and the hazard rate function (hrf),
denoted by (; , ), of the TNDL are given for = 0,1,2, … , > 0 and
∈ (0,1) by
(1 + ) + (1 − + )

(; , ) = ( ≥ ) = ()
+
and
2
[1 + (1 + )]
(; , ) = .
(1 + ) + (1 − + )

Without any loss of generality, our derivations shall focus on of the single
parameter natural discrete Lindley (NDL) distribution, i.e., TNDL when = 1.
We note that the NDL distribution is the counterpart of the single parameter
continuous Lindley distribution.

2. The NDL distribution
Definition 2
Let be a non-negative random variable obtained as a finite mixture of
geometric () and negative binomial (2, ) with mixing probabilities and
1 ,respectively. The new distributions specified by the pmf +1
+1
2

(; ) = (2 + )(1 − ) , = 0,1,2, … ∈ (0,1). (1)
1 +
The corresponding sf of the NDL distribution is given by

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