*2.2. Statistical Properties of the BNDL Distribution*

Primarily in this section, we provide some explicit results based on the mathematical properties of the BNDL distribution.

#### 2.2.1. Moment-Generating Function

If *X* ∼ BNDL(*p*) distribution, then the moment-generating function of *X* is given as

$$M\_X(t) = E\left(e^{tX}\right) = \sum\_{x=0}^{\infty} e^{tx} \frac{\left(1 - p\right)^x \left(1 + x + 2p - p^2\right)}{\left(p + 1\right)\left(2 - p\right)^{x + 2}} = \frac{1 - p\left(e^t - 2\right) + p^2\left(e^t - 1\right)}{\left(2 - e^t + p e^t - p\right)^2 \left(p + 1\right)}.$$

For more on generating functions, see Yalcin and Simsek [17], Yalcin and Simsek [18] and Simsek [19].

**Figure 1.** Pmf of BNDL distribution for some choices of *p*.
