Characterization of Life Distributions Using Log-odds Rate in Discrete Aging

Generating discrete analogues of continuous probability distributions-A survey of methods and constructions

“…Englehardht and Li (2011) applied this distribution in modeling microbial counts. See also Bakouch et al (2012) and Khorashiadizadeh et al (2012) for applications.…”

“…It can arises as a limiting q-hyper-Poisson-I (Kemp 2002) distribution and also as a mixture of Heine distributions (Benkherouf and Bather 1988). Khorashiadizadeh et al (2012) referred this distribution as discrete truncated normal. For an approximation result on this distribution see Byers and Shenton (1994).…”

Is there an absolutely continuous random variable with equal probability density and cumulative distribution functions in its support? Is it unique? What about in the discrete case?

“…In the same way as in the continuous case, in the discrete case there is a family of location parametric discrete random variables whose probability in each point of its support is equal to the probability accumulated up to that point 1 , being the uniqueness conditional to the location parameter. The discrete random variable   with mass function defined as:…”

“…As an example, Figure 1 shows the density (and cumulative) function [1] for  = 3. The parameter  sets the upper limit for the support of .…”

“…For instance, in a survival context or in reliability engineering (see, e.g., Khorashadizadeh et al [1]), where a controlled event has been detected at moment  and therefore it is known that it has occurred at an instant t≤, the parameter  graduates the risk of (actual) occurrence of the controlled event at previous instants. Larger values of  express higher risks of occurrence close to , the detection instant.…”