**Appendix A**

Density function of the gamma, exponential and Weibull distributions, respectively, are given by *Symmetry* **2019**, *11*, 1150

> Gamma distribution:

$$f(\mathbf{x}; \alpha, \beta) = \frac{\beta^{\alpha}}{\Gamma(\alpha)} \mathbf{x}^{\alpha - 1} e^{-\beta \mathbf{x}}.$$

with, *x* > 0, *α* > 0 and *β* > 0. Exponential distribution:

$$f(x; \beta) = \frac{1}{\beta}e^{-x/\beta},$$

with, *x* > 0 and *β* > 0. Weibull distribution:

$$f(\mathbf{x}; \gamma, \boldsymbol{\beta}) = \frac{\gamma}{\beta} \mathbf{x}^{\gamma - 1} e^{-\mathbf{x}^{\gamma}/\beta} \boldsymbol{\gamma}$$

with, *x* > 0, *γ* > 0 and *β* > 0.

In the following, Lemma presents an important result used in the derivation of the pdf for the MSHN distribution.

**Lemma A1.** *Prudnikov et al. [18], Equation (2.3.1.13) For γ* > 0 *, a* > 0*, r* > 0 *and s* > 0*. Then*

$$\int\_0^\infty \mathbf{x}^{\gamma - 1} \exp \left( -a\mathbf{x}^r - s\mathbf{x} \right) d\mathbf{x} = N(\gamma, a, r, \mathbf{s}), \tag{A1}$$

*where*

$$N = \begin{cases} \sum\_{j=0}^{q-1} \frac{(-a)^j}{j! s^{\gamma+r} \Gamma(\gamma+r)} \Gamma(\gamma+r)\_{p+1} \,\_{\mathbb{P}} \mathbf{F}\_q(1, \Delta(p, \gamma+r); \Delta(q, 1+j); (-1)^q z), & \text{if } 0 < r < 1 \\\sum\_{h=0}^{p-1} \frac{(-s)^h}{r! h! a^{(\gamma+h)}/r} \Gamma(\frac{\gamma+h}{r})\_{q+1} \,\_{\mathbb{P}} \mathbf{F}\_l(1, \Delta(q, \frac{\gamma+h}{r}); \Delta(p, 1+h); \frac{(-1)^p}{z}), & \text{if } r > 1 \\\frac{\Gamma(\gamma)}{(r+s)^{\gamma}} & \text{if } r = 1, \end{cases}$$

considering *γ* = *p*/*q*, *p* ≥ 1 and *q* ≥ 1 are coprime integers, where *z* = ( *p s* )*p*( *a q* )*q* , <sup>Δ</sup>(*k*, *a*) = *a k* , (*a*+<sup>1</sup>) *k* ,..., (*a*+*k*−<sup>1</sup>) *k* and *<sup>p</sup>Fq*(., ., .) is the generalized hypergeometric function defined by

$${}\_{p}F\_{q}(a\_{1},\ldots,a\_{p};b\_{1},\ldots,b\_{q};\mathtt{x}) = \sum\_{k=0}^{\infty} \frac{(a\_{1})\_{k}(a\_{2})\_{k}\ldots(a\_{p})\_{k}\mathtt{x}^{k}}{(b\_{1})\_{k}(b\_{2})\_{k}\ldots(b\_{p})\_{k}k!}$$

where (*c*)*k* = *c*(*c* + <sup>1</sup>)...(*<sup>c</sup>* + *k* − <sup>1</sup>).
