**3. Weibull distribution**

The Weibull distribution was used in 1951 by researcher Waldi Weibull in many experiments related to the reliability in the mechanical aspect and survival in the human aspect.

The emergence of this distribution, especially in the Second World War, and its wide applications in the field of reliability and life tests, was the focus of the attention of a number of researchers in this field, great in theory and practice.

The probability density function (pdf) for a two-parameter Weibull distribution is in the following form [4]:

$$\mathbf{f}(\mathbf{t}) = \frac{\mathfrak{d}}{\Theta} \mathbf{t}^{\beta - 1} \mathbf{e}^{-\frac{\vartheta}{\theta}}; \theta > 0, t > 0 \tag{1}$$

Since: β: shape parameter.

θ: Scale parameter.

The formula for the (CDF) cumulative function is:

$$\mathbf{F(t)} = \mathbf{1} - \mathbf{e^{-\frac{t^0}{\theta}}} \tag{2}$$

The formula for the survival function is:

$$\mathbf{S(t) = e^{-\frac{t^{\beta}}{\theta}}} \tag{3}$$

and the formula for moment *rth* is:

$$\mathbf{M}\_{\mathbf{r}} = \boldsymbol{\Theta}^{\mathsf{f}} \mathbf{r} \left(\mathbf{1} + \frac{\mathbf{r}}{\mathsf{\beta}}\right) \tag{4}$$
