Characterizations of Twenty (2020-2021) Proposed Discrete Distributions

In this paper, certain characterizations of twenty newly proposed discrete distributions: the discrete generalized Lindley distribution of El-Morshedy et al.(2021), the discrete Gumbel distribution of Chakraborty et al.(2020), the skewed geometric distribution of Ong et al.(2020), the discrete Poisson X gamma distribution of Para et al.(2020), the discrete Cos-Poisson distribution of Bakouch et al.(2021), the size biased Poisson Ailamujia distribution of Dar and Para(2021), the generalized Hermite-Genocchi distribution of El-Desouky et al.(2021), the Poisson quasi-xgamma distribution of Altun et al.(2021a), the exponentiated discrete inverse Rayleigh distribution of Mashhadzadeh and MirMostafaee(2020), the Mlynar distribution of Frühwirth et al.(2021), the flexible one-parameter discrete distribution of Eliwa and El-Morshedy(2021), the two-parameter discrete Perks distribution of Tyagi et al.(2020), the discrete Weibull G family distribution of Ibrahim et al.(2021), the discrete Marshall–Olkin Lomax distribution of Ibrahim and Almetwally(2021), the two-parameter exponentiated discrete Lindley distribution of El-Morshedy et al.(2019), the natural discrete one-parameter polynomial exponential distribution of Mukherjee et al.(2020), the zero-truncated discrete Akash distribution of Sium and Shanker(2020), the two-parameter quasi Poisson-Aradhana distribution of Shanker and Shukla(2020), the zero-truncated Poisson-Ishita distribution of Shukla et al.(2020) and the Poisson-Shukla distribution of Shukla and Shanker(2020) are presented to complete, in some way, the authors’ works.


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. Therefore, the problem of characterizing a distribution is an important problem in applied sciences, where an investigator is vitally interested to know if their model follows the right distribution. To this end, the investigator relies on conditions under which their model would follow specifically the chosen distribution. El In this paper, we present two characterizations of these distributions based on: (i) the conditional expectation of certain function of the random variable; (ii) the hazard rate functions of DsGLi, P-Xgamma, SBPA, GHGD, PQX, EDIR, Mlynar, DsFx-I, DWG, DMOL and ZTDAD and (iii) the reverse hazard functions of DGu, SG, CosPois, DP, EDLi, NDOPPE, QPAD, ZTPID and PSD distributions.
Proof. If X has pmf (2), then the left-hand side of (61) will be Conversely, if (61) holds, then where we have used F (k) = F (k + 1) − f (k + 1). From (61), we also have Now, subtracting (63) from (62), we arrive at From the last equality, after some computations, we arrive at which, in view of (3), implies that X has pmf (2). Proposition 2.1.2. Let X : Ω → Z be a random variable. The pmf of X is (5) if and only if Proof. If X has pmf (5), then the left-hand side of (64) , using the telescoping series property, will be Now, subtracting (65) from (66), we arrive at From the last equality, we have which, in view of (6), implies that X has pmf (5).
Proof. If X has pmf (8), then the left-hand side of (67) , using the telescoping series property, will be From (67), we also have Now, subtracting (68) from (69), we arrive at From the last equality, after a good deal of simplifications, we have which, in view of (9), implies that X has pmf (8).
Characterizations of Twenty (2020-2021) Proposed Discrete Distributions Proposition 2.1.4. Let X : Ω → N * be a random variable. The pmf of X is (11) if and only if Proof. If X has pmf (11), then the left-hand side of (70) , using the infinite geometric series property, will be Conversely, if (70) holds, then Now, subtracting (72) from (71), we arrive at From the last equality, after a good deal of algebra, we have which, in view of (12), implies that X has pmf (11). Proposition 2.1.5. Let X : Ω → N * be a random variable. The pmf of X is (14) if and only if Characterizations of Twenty (2020-2021) Proposed Discrete Distributions Proof. If X has pmf (14), then the left-hand side of (73) , using the finite geometric series property, will be Conversely, if (73) holds, then From (73), we also have Now, subtracting (74) from (75), we arrive at From the last equality, after a good deal of simplifications, we have which, in view of (15), implies that X has pmf (14).
Characterizations of Twenty (2020-2021) Proposed Discrete Distributions Proof. If X has pmf (17), then the left-hand side of (76) will be From (76), we also have Now, subtracting (78) from (77), we have From the last equality, after some computations, we arrive at which, in view of (18), implies that X has pmf (17). Proposition 2.1.7. Let X : Ω → N * be a random variable. The pmf of X is (20) if and only if Proof. If X has pmf (20), then the left-hand side of (79) will be .

Characterizations of Twenty (2020-2021) Proposed Discrete Distributions
Conversely, if (79) holds, then From (79), we also have Now, subtracting (81) from (80), we have From the last equality, after some computations, we arrive at which, in view of (21), implies that X has pmf (20). Proposition 2.1.8. Let X : Ω → N * be a random variable. The pmf of X is (23) if and only if Proof. If X has pmf (23), then for k ∈ N * , the left-hand side of (82) , using infinite geometric series formula, will be Characterizations of Twenty (2020-2021) Proposed Discrete Distributions Conversely, if (82) holds, then From (82), we also have Now, subtracting (84) from (83), yields From the above equality, after some regrouping the terms and simplifications, we have which, in view of (24), implies that X has pmf (23). Proposition 2.1.9. Let X : Ω → N * be a random variable. The pmf of X is (26) if and only if Proof. If X has pmf (26), then the left-hand side of (85) will be From (85), we also have Now, subtracting (87) from (86), we arrive at From the last equality, after some computations, we arrive at which, in view of (27), implies that X has pmf (26).
Proof. If X has pmf (29), then the left-hand side of (88), using finite geometric sum formula, will be Conversely, if (88) holds, then From (88), we also have Characterizations of Twenty (2020-2021) Proposed Discrete Distributions Now, subtracting (90) from (89), we arrive at From the last equality, after some computations, we arrive at which, in view of (30), implies that X has pmf (29). Proposition 2.1.11. Let X : Ω → N * be a random variable. The pmf of X is (32) if and only if Proof. If X has pmf (32), then the left-hand side of (91), using finite geometric sum formula, will be Conversely, if (91) holds, then From (91), we also have Characterizations of Twenty (2020-2021) Proposed Discrete Distributions Now, subtracting (93) from (92), we arrive at From the last equality, after some computations, we arrive at which, in view of (33), implies that X has pmf (32). Proposition 2.1.12. Let X : Ω → N * be a random variable. The pmf of X is (35) if and only if Proof. If X has pmf (35), then the left-hand side of (94) , using finite geometric sum, will be (F (k)) Conversely, if (94) holds, then From (94), we also have Characterizations of Twenty (2020-2021) Proposed Discrete Distributions Now, subtracting (95) from (96), we arrive at From the last equality, we have which, in view of (36), implies that X has pmf (35). Proposition 2.1.13. Let X : Ω → N * be a random variable. The pmf of X is (38) if and only if Proof. If X has pmf (38), then the left-hand side of (97) , using telescoping sum, will be

Conversely, if (97) holds, then
Characterizations of Twenty (2020-2021) Proposed Discrete Distributions From (97), we also have Now, subtracting (99) from (98), we arrive at From the last equality, we have which, in view of (39), implies that X has pmf (38). Proposition 2.1.14. Let X : Ω → N * be a random variable. The pmf of X is (41) if and only if Proof. If X has pmf (41), then the left-hand side of (100) , using telescoping sum, will be Conversely, if (100) holds, then Characterizations of Twenty (2020-2021) Proposed Discrete Distributions From (100), we also have Now, subtracting (102) from (101), we arrive at From the last equality, we have which, in view of (42), implies that X has pmf (41).
Proof. If X has pmf (44), then the left-hand side of (103) , using telescoping sum, will be Conversely, if (103) holds, then From (103), we also have From the last equality, we have which, in view of (45), implies that X has pmf (44). Proposition 2.1.16. Let X : Ω → N be a random variable. The pmf of X is (47) if and only if Proof. If X has pmf (47), then the left-hand side of (106) , using finite geometric sum, will be Conversely, if (106) holds, then From (106), we also have Now, subtracting (107) from (108), we arrive at From the last equality, after some computations, we have which, in view of (48), implies that X has pmf (47). Proposition 2.1.17. Let X : Ω → N be a random variable. The pmf of X is (50) if and only if Characterizations of Twenty (2020-2021) Proposed Discrete Distributions Proof. If X has pmf (50), then the left-hand side of (109) , using infinite geometric sum, will be Conversely, if (109) holds, then From (109), we also have Now, subtracting (111) from (110), we arrive at From the last equality, after some computations, we have which, in view of (51), implies that X has pmf (50). Proposition 2.1.18. Let X : Ω → N * be a random variable. The pmf of X is (53) if and only if Characterizations of Twenty (2020-2021) Proposed Discrete Distributions Proof. If X has pmf (53), then the left-hand side of (112) , using finite geometric sum, will be Conversely, if (112) holds, then From (112), we also have Now, subtracting (113) from (114), we arrive at From the last equality, after some computations, we have which, in view of (54), implies that X has pmf (53). Proposition 2.1.19. Let X : Ω → N be a random variable. The pmf of X is (56) if and only if Characterizations of Twenty (2020-2021) Proposed Discrete Distributions Proof. If X has pmf (56), then the left-hand side of (115) , using finite geometric sum, will be Conversely, if (115) holds, then From (115), we also have Now, subtracting (116) from (117), we arrive at From the last equality, after some computations, we have which, in view of (57), implies that X has pmf (56). Proposition 2.1.20. Let X : Ω → N be a random variable. The pmf of X is (57) if and only if Characterizations of Twenty (2020-2021) Proposed Discrete Distributions Proof. If X has pmf (59), then the left-hand side of (118) , using finite geometric sum, will be Conversely, if (118) holds, then From (118), we also have Now, subtracting (119) from (120), we arrive at From the last equality, after some computations, we have which, in view of (60), implies that X has pmf (59).

Characterizations of
with the initial condition h Proof. If X has pmf (2), then clearly (121) holds. Now, if (121) holds, then for every x ∈ N * , we have In which, in view of (3), implies that X has pmf (2). Proposition 2.2.2. Let X : Ω → N * be a random variable. The pmf of X is (11) if and only if its hazard rate function satisfies the difference equation k ∈ N * , with the boundary condition h F (0) = 2γ −1 θ 2 1 + 4θ + θ 2 .
In view of the fact that h F (0) = θ 2 [2(1+θ) 2 +2θ] γ , from the last equation we have which, in view of (12), implies that X has pmf (11). Proposition 2.2.3. Let X : Ω → N be a random variable. The pmf of X is (17) if and only if its hazard rate function satisfies the difference equation with the initial condition h F (1) = (17), then clearly (123) holds. Now, if (123) holds, then for every x ∈ N, we have In view of the fact that h F (1) = which, in view of (18), implies that X has pmf (17). Proposition 2.2.4. Let X : Ω → N * be a random variable. The pmf of X is (20) if and only if its hazard rate function satisfies the difference equation with the initial condition h F (0) = 1 α Hn,m(γ,β) G(α;β,γ+1) .

So, we have
.