Characterizations of Three (2020) Introduced Discrete Distributions

The problem of characterizing a probability distribution is an important problem which has attracted the attention of many researchers in the recent years. To understand the behavior of the data obtained through a given process, we need to be able to describe this behavior via its approximate probability law. This, however, requires to establish conditions which govern the required probability law. In other words we need to have certain conditions under which we may be able to recover the probability law of the data. So, characterization of a distribution plays an important role in applied sciences, where an investigator is vitally interested to find out if their model follows the selected distribution. In this short note, certain characterizations of three recently introduced discrete distributions are presented to complete, in some way, the works of Hussain(2020), Eliwa et al.(2020) and Hassan et al.(2020).


Introdution
To understand the behavior of the data obtained through a given process, we need to be able to describe this behavior via its approximate probability law. This, however, requires to establish conditions which govern the required probability law. In other words we need to have certain conditions under which we may be able to recover the probability law of the data. So, characterization of a distribution is important in applied sciences, where an investigator is vitally interested to find out if their model follows the selected distribution. Therefore, the investigator relies on conditions under which their model would follow a specified distribution. A probability distribution can be characterized in different directions one of which is based on the truncated (or conditional) moments. This type of characterization initiated by Galambos and Kotz(1978) and followed by other authors such as Kotz and Shanbhag(1980), Glänzel et al.(1984), Glänzel(1987), Glänzel and Hamedani(2001) and Kim and Jeon(2013), to name a few. For example, Kim and Jeon(2013) proposed a credibility theory based on the truncation of the loss data to estimate conditional mean loss for a given risk function. It should also be mentioned that characterization results are mathematically challenging and elegant. Hussain(2020) introduced a new discrete probability model called Zero Truncated Discrete Lindley (ZTDL) distribution, Eliwa et al.(2020) proposed a discrete version of the Gompertz-G distribution called Discrete Gompertz-G (DG z -G) distribution and Hassan et al.(2020) introduced a new flexible discrete distribution called Poisson Aliamujia (PA) distribution. We intend to present certain characterizations of these distributions to complete, in some way, the works mentioned above. These characterizations presented below are based on (i) the conditional expectation of certain function of the random variable and (ii) the hazard function.
The cumulative distribution function (cdf), F (x), the corresponding probability mass function (pmf), f (x), and the hazard function, h F (x), of each of the distributions ZTDL, DG z -G and PA are given, respectively, by where β > 0 and p ∈ (0, 1) are parameters; where c > 0 and p ∈ (0, 1) are parameters; and f (x) = 4α 2 (1 + x) where α > 0 is a parameter.  (3), (6) and (9) have been rewritten for the sake of the simplicity of the related computations.

Characterization results
We present our characterizations (i) and (ii) in the following two sub-sections.

Characterization based on the conditional expectation of a function of the random variable
In this sub-section, we use the fact that the hazard function uniquely determines the distribution of a random variable (see Nair et al.(2018)).
) be a random variable and let ϕ (X) = (1+2α) 2 1+X . The pmf of X is (8) if and only if the conditional expectation of ϕ (X) given X > k, is of the form Proof. If X has pmf (8), then the left-hand side of (16) will be Conversely, if (16) holds, then From (17), we also have ∞