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<h3>Functional relationships for <script type="math/tex">\;X \sim GENG(\theta,\beta,\kappa),\;\;T \in [0,\infty),\;\;\; \theta,\beta,\kappa\; > 0</script>
</h3>

$$
\begin{aligned}
f(t|\theta,\beta,\kappa)&=\Gamma_{I}\left[\left(\frac{t}{\theta}\right)^{\beta},\kappa\right]\\\\\\\\
F(t|\theta,\beta,\kappa)&=\frac{\beta}{\Gamma(\kappa)\theta}\left(\frac{t}{\theta}\right)^{\kappa\beta-1}\exp\left[-\left(\frac{t}{\theta}\right)^{\beta}\right]\\\\\\\\
t_{p}&=\theta \left[\Gamma_{I}^{-1}(p;\kappa)\right]^{1/\beta}\\\\\\\\
E[T]&=\frac{\theta\;\Gamma(1/\beta)+\kappa}{\Gamma(\kappa)}\\\\\\\\
Var[T]&=\theta^2\left[\frac{\Gamma(2/\beta+\kappa)}{\Gamma(\kappa)}-\frac{\Gamma^{2}(1/\beta)+\kappa}{\Gamma^{2}(\kappa)}\right]
\end{aligned}
$$

- $\theta \in \mathbb{R}^{+}$ is a scale parameter

- $\beta \in \mathbb{R}^{+}$ is a shape parameter

- $\kappa \in \mathbb{R}^{+}$ is a shape parameter
    
- $\kappa \in \mathbb{R}^{+}$ is a shape parameter
    
- $\theta \in \mathbb{R}^{+}$ is a scale parameter

- $\Gamma(z)$ is the gamma function defined as 

$$
\Gamma(z) = 
\begin{cases}
\int_0^{\infty} x^{z-1}e^{-x}dx \hspace{12pt}\text{ if } z \in \mathbb{R}\\\\
(z - 1)! \hspace{40pt} \mbox{ if } z \in \mathbb{I}
\end{cases}
$$

- $\Gamma_{I}(a,b)$ is the lower incomplete gamma function defined as

$$
\Gamma_{I}(a,b) = \int_{0}^{b} t^{a-1}e^{-t}dt.
$$

- $\Gamma_{I}^{-1}(a,b)$ is the inverse of the lower incomplete gamma function defined as

$$
\Gamma_{I}^{-1}(a,b) = \frac{\Gamma_{I}(b,a)}{\Gamma(b)} = \frac{1}{\Gamma(b)}\int_{0}^{a} t^{b-1}e^{-t}dt.
$$

- $\Gamma(z)$ values can be computed in R using the base function `gamma(x)` 

- $\Gamma_{I}(a,b)$ values can be computed using the `gamma_inc(a,b)` function from the `gsl` package

- $\Gamma_{I}^{-1}(a,b)$ values can be computed using the `gamma_inc_P(a,b)` function from the `gsl` package