One parameter continuous Shanker distribution introduced by Shanker¹¹ with parameter  is defined by its probability density function (pdf)

$f\left(x:\theta \right)= \frac{{\theta }^2}{{\theta }^2+1} \left(\theta +x\right) e^{-\theta x}.x>0. \theta >0.$                                         (2.1)

Discretization of continuous distribution can be done using different methodologies. In this paper we deal with the derivation of a new discrete distribution which may be called discrete Shanker (DS) distribution. It takes values in {0, 1, 2, . . .,}. This distribution is generated by discretizing the survival function of the continuous Shanker distribution

$S\left(x\right)=\underset{x}{\overset{\infty }{\int }}f\left(x:\theta \right)dx$

$= \frac{{\theta }^2+1+ \theta x}{{\theta }^2+1} e^{-\theta x},x>0.\theta >0.$                                                 (2.2)

$S\left(x+1\right)= \frac{{\theta }^2+1+ \theta \left(x+1\right)}{{\theta }^2+1} e^{-\theta \left(x+1\right)},x>0.\theta >0.$                       (2.3)

Where $f\left(x;\theta \right)$ denotes the pdf of Shanker distribution.

The pmf of discrete Shanker distribution $f_D\left(x;\theta \right)$  may be obtained as

$f_D\left(x;\theta \right)=S\left(x\right)-S\left(x+1\right)$

= $\frac{({\theta }^2+1+ \theta x)\left(1- e^{-\theta }\right)-\theta e^{-\theta }}{{\theta }^2+1} e^{-\theta x},x=0,1,2,\mathrm{3....}$                      (2.4)

Proposition 1: The probability generating function (pgf) of DS distribution is given by

$G_D\left(t\right)=\frac{\left({\theta }^2+1\right)\left(1- e^{-\theta }\right)-\theta }{\left({\theta }^2+1\right)\left(1- e^{-\theta }t\right)}+\frac{\theta \left(1- e^{-\theta }\right)}{\left({\theta }^2+1\right){\left(1-e^{-\theta }t\right)}^2}$

Proposition 2: The cumulative distribution of DS distribution is given by

$F\left(x\right)=\frac{\left({\theta }^2+1\right)-\left({\theta }^2+1+\theta \left(x+1\right)\right)e^{-\theta \left(x+1\right)}}{\left({\theta }^2+1\right)}$

The survival function of DS distribution has obtained as

$S_D\left(x\right)=\frac{\left({\theta }^2+1+\theta \left(x+1\right)\right)e^{-\theta \left(x+1\right)}}{\left({\theta }^2+1\right)}$

The failure hazard rate may be obtained as

$r_D\left(x\right)=\frac{({\theta }^2+1+ \theta x)\left(1- e^{-\theta }\right)-\theta e^{-\theta }}{\left({\theta }^2+1+\theta x\right)}$

The reversed failure rate

$r^{*}\left(x\right)=\frac{\left[({\theta }^2+1+ \theta x)\left(1- e^{-\theta }\right)-\theta e^{-\theta }\right]e^{-\theta x}}{\left({\theta }^2+1\right)-\left({\theta }^2+1+\theta \left(x+1\right)\right)e^{-\theta \left(x+1\right)}}$

The second rate of failure is obtained as

$r^{*}\left(x\right)=\log \left[\frac{s\left(x\right)}{s\left(x+1\right)}\right]$

$=\log \left[\frac{\left({\theta }^2+1+\theta \left(x+1\right)\right)}{e^{-\theta }\left({\theta }^2+1+\theta \left(x+2\right)\right)}\right]$

The Proportions of probabilities is given by

$\frac{f_D\left(x+1;\theta \right)}{f_D\left(x;\theta \right)}=e^{-\theta }\left[1+\frac{\theta \left(1- e^{-\theta }\right)}{({\theta }^2+1+ \theta x)\left(1- e^{-\theta }\right)-\theta e^{-\theta }}\right]$

Probability recurrence relation:

 Probability recurrence relation of DS distribution may be obtained as

$P_{r+2}= e^{-\theta }\left(2P_{r+1}- e^{-\theta }{P}_r\right)\text{r}=1,2,3,\dots$                                                                                (2.5)

Where $P_0=\frac{({\theta }^2+1)\left(1- e^{-\theta }\right)-\theta e^{-\theta }}{{\theta }^2+1}$ , and

$P_1=\frac{({\theta }^2+1+ \theta )\left(1- e^{-\theta }\right)-\theta e^{-\theta }}{{\theta }^2+1} e^{-\theta }$                                                                   (2.6)

 Here $P_r$  denotes Pr(X= r).

Factorial moment recurrence relation

Factorial moment generating function (fmgf) may be obtained as

$MD\left(t\right)=$ $\frac{\left({\theta }^2+1\right)\left(1- e^{-\theta }\right)-\theta }{\left({\theta }^2+1\right)\left(1-e^{-\theta }- e^{-\theta }t\right)}+\frac{\theta \left(1- e^{-\theta }\right)}{\left({\theta }^2+1\right){\left(1-e^{-\theta }- e^{-\theta }t\right)}^2}$ .                                                     (2.7)

First four factorial moments may be obtained as

${\mu }_{\left[1\right]}^{\text{'}}=\frac{ e^{-\theta }\left[\left({\theta }^2+1\right)\left(1-e^{-\theta }\right)+\theta \right]}{\left({\theta }^2+1\right){\left(1-e^{-\theta }\right)}^2}$

${\mu }_{\left[2\right]}^{\text{'}}=\frac{2 e^{-2\theta }\left[\left({\theta }^2+1\right)\left(1-e^{-\theta }\right)+2\theta \right]}{\left({\theta }^2+1\right){\left(1-e^{-\theta }\right)}^3}$

${\mu }_{\left[3\right]}^{\text{'}}=\frac{6 e^{-3\theta }\left[\left({\theta }^2+1\right)\left(1-e^{-\theta }\right)+3\theta \right]}{\left({\theta }^2+1\right){\left(1-e^{-\theta }\right)}^4}$

${\mu }_{\left[4\right]}^{\text{'}}=\frac{24 e^{-4\theta }\left[\left({\theta }^2+1\right)\left(1-e^{-\theta }\right)+4\theta \right]}{\left({\theta }^2+1\right){\left(1-e^{-\theta }\right)}^4}$

Proposition 3: The general form of factorial moment may also be written as

${\mu }_{\left[r\right]}^{\text{'}}=\frac{r! e^{-\theta r}\left[\left({\theta }^2+1\right)\left(1-e^{-\theta }\right)+\theta r\right]}{\left({\theta }^2+1\right){\left(1-e^{-\theta }\right)}^{r+1}}$                                                      (2.8)

Hence, mean and variance may be obtained as

Mean $=\frac{ e^{-\theta }\left[\left({\theta }^2+1\right)\left(1-e^{-\theta }\right)+\theta \right]}{\left({\theta }^2+1\right){\left(1-e^{-\theta }\right)}^2}$ , and

$Variance=\frac{ e^{-\theta }\left[{\left({\theta }^2+1\right)}^2{\left(1-e^{-\theta }\right)}^2+\left({\theta }^2+1\right)\left(1-e^{-\theta }\right)\theta -e^{-\theta }{\theta }^2\right]}{{\left({\theta }^2+1\right)}^2{\left(1-e^{-\theta }\right)}^4}$  respectively.

Zero- truncated distributions are applicable for the situations when the data to be modeled originate from a generating mechanism that structurally excludes zero counts. The discrete Shanker distribution must be adjusted to count for the missing zeros. Here the zero-truncated discrete Shanker distribution has been derived.

 The pmf $f_z\left(x;\theta \right)$ of Zero-truncated DS distribution has been derived as

$f_z\left(x;\theta \right)=\frac{P_x}{1-P_0}$                                                                                                                 (3.0)

Where $P_x$ denotes the pmf of discrete Shanker distribution.

$Hence,f_z\left(x;\theta \right)=\frac{({\theta }^2+1+ \theta x)\left(1- e^{-\theta }\right)-\theta e^{-\theta }}{({\theta }^2+\theta +1)} e^{-\theta \left(x-1\right)},x=1, 2,3,\dots$                           (3.1)

Probability recurrence relation for ZTDS distribution

The pgf $G_z\left(t\right)$ of zero-truncated DS distribution may be obtained as

$G_z\left(t\right)={{\displaystyle {\sum }_{x=1}^{\infty }t}}^x{f}_z\left(x;\theta \right),$

$=\frac{t\left[\left\{\left({\theta }^2+1\right)\left(1-e^{-\theta }\right)-\theta e^{-\theta }\right\}\left(1-t{e}^{-\theta }\right)+\theta \left(1-e^{-\theta }\right) \right]}{\left(({\theta }^2+\theta +1\right){\left(1-t{e}^{-\theta }\right)}^2}$                                                      (3.2)

Probability recurrence relation ZTDS distribution may obtained as

$P^z{}_r=e^{-\theta }\left[2P^z{}_{r-1}-e^{-\theta }{P}^z{}_{r-2}\right]r=2,3,4,\dots .$

Where $P^z{}_1=\frac{({\theta }^2+1+ \theta )\left(1- e^{-\theta }\right)-\theta e^{-\theta }}{({\theta }^2+\theta +1)}$ and $P^z{}_2=\frac{({\theta }^2+1+2\theta )\left(1- e^{-\theta }\right)-\theta e^{-\theta }}{({\theta }^2+\theta +1)} e^{-\theta }$                 (3.3)

Proposition 4: The cumulative distribution of ZTDS distribution is given by

$F_z\left(x\right)=\frac{\left({\theta }^2+\theta +1\right)-\left({\theta }^2+\theta +\theta x+1\right)e^{-\theta x}}{\left({\theta }^2+\theta +1\right)}$

The survival function of ZTDS distribution is given by

$S_z\left(x\right)=\frac{\left({\theta }^2+\theta +\theta x+1\right))e^{-\theta x}}{\left({\theta }^2+\theta +1\right)}$

The Failure hazared rate may be obtained as

$r_z\left(x\right)=\frac{P\left(X=x\right)}{P\left(X\ge x-1\right)}$ ,

$=\frac{({\theta }^2+1+ \theta x)\left(1- e^{-\theta }\right)-\theta e^{-\theta }}{\left({\theta }^2+1+\theta x\right)}$ .

The reversed failure rate

$r_z^{*}\left(x\right)=\frac{P\left(X=x\right)}{P\left(X\le x\right)}$

$=\frac{({\theta }^2+1+ \theta x)\left(1- e^{-\theta }\right)-\theta e^{-\theta }}{\left({\theta }^2+1+\theta x\right)}$ .

The second rate of failure is obtained as

$r_z^{*}\left(x\right)=\frac{P\left(X=x\right)}{P\left(X\le x\right)}$

$=\frac{[({\theta }^2+1+ \theta x)\left(1- e^{-\theta }\right)-\theta e^{-\theta }]e^{-\theta \left(x-1\right)}}{\left({\theta }^2+\theta +1\right)-\left({\theta }^2+\theta +1+\theta x\right))e^{-\theta x}}$

The proportions of probabilities is given by

$\frac{f_z\left(x+1;\theta \right)}{f_z\left(x;\theta \right)}=e^{-\theta }\left[1+\frac{\theta \left(1- e^{-\theta }\right)}{({\theta }^2+1+ \theta x)\left(1- e^{-\theta }\right)-\theta e^{-\theta }}\right]$

Factorial moment recurrence relation for ZTDS distribution 

Factorial moment generating function $M_z\left(t\right)$  of ZTDS distribution may be obtained as

$M_z\left(t\right)=\frac{\left(1+t\right)\left[\left\{\left({\theta }^2+1\right)\left(1-e^{-\theta }\right)-\theta e^{-\theta }\right\}\left(1-t-t{e}^{-\theta }\right)+\theta \left(1-e^{-\theta }\right) \right]}{\left(({\theta }^2+\theta +1\right){\left(1-t-t{e}^{-\theta }\right)}^2}$                                      (3.4)

Factorial moment recurrence relation of ZTDS distribution may be obtained as

${\eta }_{\left[r\right]}^{\text{'}}=\frac{e^{-\theta }}{{\left(1-e^{-\theta }\right)}^2}\left[2\left(1-e^{-\theta }\right)r-e^{-\theta }{\eta }_{\left[r-1\right]}^{\text{'}}-r\left(r-1\right)-e^{-\theta }{\eta }_{\left[r-2\right]}^{\text{'}}\right]$ , $r\ge 2$                     (3.5)

where $\eta$

${\eta }_{\left[1\right]}^{\text{'}}=\frac{ \left[\left({\theta }^2+1\right)\left(1-e^{-\theta }\right)+\theta \right]}{\left({\theta }^2+\theta +1\right){\left(1-e^{-\theta }\right)}^2}$

${\eta }_{\left[2\right]}^{\text{'}}=\frac{ 2e^{-\theta }\left[\left({\theta }^2+1\right)\left(1-e^{-\theta }\right)+2\theta \right]}{\left({\theta }^2+\theta +1\right){\left(1-e^{-\theta }\right)}^3}$                  (3.6)

Variance ${\sigma }_Z{}^2$  of ZTDS distribution may be obtained as

${\sigma }_Z{}^2=\frac{ \left[{\left({\theta }^2+1\right)}^2{\left(1-e^{-\theta }\right)}^2+5\left({\theta }^2+1\right)\left(1-e^{-\theta }\right)e^{-\theta }\theta +4{\theta }^2{e}^{-\theta }\right]}{{\left({\theta }^2+\theta +1\right)}^2{\left(1-e^{-\theta }\right)}^4}$

Proposition 5: The general form of factorial moment may be written as

${\eta }_{\left[r\right]}^{\text{'}}=\frac{r! e^{-\theta \left(r-1\right)}\left[\left({\theta }^2+1\right)\left(1-e^{-\theta }\right)+\theta r\right]}{\left({\theta }^2+\theta +1\right){\left(1-e^{-\theta }\right)}^{r+1}}$ .                       (3.7)