Discretization of continuous distribution can be done using different methodologies. In this paper we deal with the derivation of a new discrete distribution which takes values in $\left\{0,1,...\right\}$ . This new distribution is generated by discretizing the continuous survival function of the Shanker distribution, which is may be obtained as

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

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

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

The probability mass function (pmf) of discrete Shanker distribution may be obtained as

$P\left(X=x\right)=S\left(x\right)-S\left(x+1\right)$
$=\frac{({\theta }^2+1+ \theta x)\left(1- e^{-\theta }\right)-\theta e^{-\theta }}{{\theta }^2+1}{e}^{(-\theta x)},$ $$x=0, 1,2, 3$$ (2.3)

Probability recurrence relation 

Probability recurrence relation of discrete Shanker distribution may be obtained as

$P_{(r+2)}=e^{-\theta }(2P_{r+1}- e^{-\theta }{P}_r),r\ge 1$ (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)

Factorial moment recurrence relation 

Factorial moment generating function (fmgf) may be obtained as

$M\left(t\right)=G\left(1+t\right)$

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

 The more 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)

If a random variable $X$  have discrete Shanker distribution with parameter $\theta$ then the pmf of the size-biased distribution may be derived as

$f_s\left(x;\theta \right)=\frac{x{P}_x}{\mu }$ , $x=1,2,3,\dots$  (3.1)

Where $P_x$ and $\mu$  denote respectively pmf and the mean of discrete Shanker distribution. 

The pmf $f_s\left(x;\theta \right)$ of size- biased discrete Shanker distribution with parameters $\theta$ may be derived from (3.1) as

$f_s\left(x,\alpha \right)=\frac{x{p}_x}{\mu }$

$=x{e}^{-\theta \left(x-1\right)}\frac{\left[({\theta }^2+1+ \theta x)\left(1- e^{-\theta }\right)-\theta e^{-\theta }\right]{\left(1-e^{-\theta }\right)}^2}{\left[\left({\theta }^2+1\right)\left(1-e^{-\theta }\right)+\theta \right]}$ $x=1,2,3,\dots$  (3.2)

Probability generating function $G_s\left(t\right)$ for Size- biased Discrete Shanker Distribution may be obtained as

$G_s\left(t\right)={\displaystyle \sum }_{x=0}^{\infty }{t}^x{P}_x$

     $=t\frac{\left[\left({\theta }^2+1\right)\left(1- e^{-\theta }\right)-\theta e^{-\theta }\right]{\left(1- e^{-\theta }\right)}^2\left(1- e^{-\theta }t\right)+\theta {\left(1- e^{-\theta }\right)}^3\left(1+e^{-\theta }t\right)}{\left[\left({\theta }^2+1\right)\left(1- e^{-\theta }\right)+\theta \right]{\left(1- e^{-\theta }t\right)}^3}$ (3.3)

Probability recurrence relation for size- biased discrete  shanker distribution

Probability recurrence relation of Size- biased Discrete Shanker Distribution distribution may be obtained as

$P_r=e^{-\theta }\left[3P_{r-1}-3e^{-\theta }{P}_{r-2}+e^{-\theta }{P}_{r-3}\right]$ for $r>2$  (3.4)

where

$P_1=\frac{\left[({\theta }^2+1+ \theta )\left(1- e^{-\theta }\right)-\theta e^{-\theta }\right]{\left(1-e^{-\theta }\right)}^2}{\left[\left({\theta }^2+1\right)\left(1-e^{-\theta }\right)+\theta \right]}$ and (3.5)

$P_2=2e^{-}\frac{\left[({\theta }^2+1+2\theta )\left(1- e^{-\theta }\right)-\theta e^{-\theta }\right]{\left(1-e^{-\theta }\right)}^2}{\left[\left({\theta }^2+1\right)\left(1-e^{-\theta }\right)+\theta \right]}$

Factorial moment recurrence relation for size- biased discrete  shanker distribution

Factorial moment generating function $M_s\left(t\right)$ of Size- biased discrete Shanker distribution may be obtained as

$M_s\left(t\right)=G_s\left(1+t\right)$

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

More general form  ${\mu }_{\left[r\right]}=\frac{r! e^{-\theta \left(r-1\right)}\left[\left({\theta }^2+1\right)\left(1-e^{-\theta }\right)\left(r+e^{-\theta }\right)+\theta \left(r^2-e^{-\theta }\right)\right]}{\left[\left({\theta }^2+1\right)\left(1- e^{-\theta }\right)+\theta \right]{\left(1-e^{-\theta }\right)}^r}$    (3.7)

Factorial moment recurrence relation of Size- biased discrete Shanker distribution may be obtained as

    ${\mu }_{\left[r\right]}^{\text{'}}=\frac{e^{-\theta }}{A^3}\left[3A^{2}r{\mu }_{\left[r-1\right]}^{\text{'}}-3Ar\left(r-1\right) e^{-\theta } {\mu }_{\left[r-2\right]}^{\text{'}}+Ar\left(r-1\right)\left(r-2\right)e^{-2\theta } {\mu }_{\left[r-3\right]}^{\text{'}}\right],$  (3.7)

Where $A= 1-e^{-\theta }.$

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

${\mu }_{\left[2\right]}=\frac{2e^{-\theta }\left[\left({\theta }^2+1\right)\left(1-e^{-\theta }\right)\left(2+e^{-\theta }\right)+\theta \left(4-e^{-\theta }\right)\right]}{\left[\left({\theta }^2+1\right)\left(1- e^{-\theta }\right)+\theta \right]{\left(1-e^{-\theta }\right)}^2}$ (3.8)

${\mu }_{\left[3\right]}=\frac{6 e^{-2\theta }\left[\left({\theta }^2+1\right)\left(1-e^{-\theta }\right)\left(3+e^{-\theta }\right)+\theta \left(9-e^{-\theta }\right)\right]}{\left[\left({\theta }^2+1\right)\left(1- e^{-\theta }\right)+\theta \right]{\left(1-e^{-\theta }\right)}^3}$

The parameter $\theta$ of Shanker distribution has been estimated using Newton’s –Raphson method by considering appropriate initial guest value for $\theta$ . The function of $\theta$ can be expressed as

Shanker et al.¹ fitted Poisson distribution (PD), Poisson- Lindley distribution (PLD) and Poisson-Akash distribution (PAD) to eleven numbers of data sets covering ecology, genetics and thunderstorms. In this investigation discrete Shanker (DS) distribution has been fitted to all 11 data sets have been considered for a comparison (Tables 1-11).^2-36

In this article, the discrete Shanker distribution has been introduced by discretizing the continuous Shanker distribution. We have studied some properties of the distributions. Further the applications of the distribution and goodness of fit of the distribution.