## 3 Materials and Methods
### Probability Distributions, Fit Methods, and Inferential Statistics  
To estimate valley streamflows and drought durations for different return periods, the authors used the probability distributions below in Table 3. These distributions include Weibull [17], Generalized Extreme Value (GEV) [18], Gumbel [19], normal [12], log-normal [20], three-parameter log-normal, gamma, generalized gamma, inverse gamma, Pearson type III [21], and log Pearson type III [20,22]. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **1970–1999** & \multicolumn{2}{c}{**2000–2021**} \\ \hline
**Year** & **Duration (Days)** & **Year** & **Duration (Days)** \\ \hline
1972 & 42 & 2000 & 40 \\
1973 & 40 & 2001 & 23 \\
1974 & 40 & 2002 & 47 \\
1975 & 26 & 2003 & 22 \\
1976 & Not available & 2004 & 21 \\
1977 & 88 & 2005 & 17  
For the fitting methods, maximum likelihood [23; 24], method of moments [12], and weighted method of moments [23] were used. Furthermore, the weighted method of moments and WRC (Water Resources Commission) [25] for the log Pearson type III probability distribution are presented in Table 4.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Probability Distribution** & **Formulation** & **Equation Number** \\ \hline Weibull & \(f(x)=\frac{\varepsilon}{a}\big{(}\frac{x}{\delta}\big{)}^{c-1}exp\left[-\big{(} \frac{x}{\delta}\big{)}^{c}\right]\) & (1) \\ \(f(x)\): Probability density function, \  
Inferential statistics serve as criteria for selecting the best-fitting probability distribution for the data under study. In this research, the authors considered the Chi-square statistic, the coefficient of variation, the kurtosis coefficient, the skewness coefficient, as well as the AIC. The Chi-square statistic is widely used to assess the existence of a relationship or discrepancy between observed and expected data in a dataset, where the recorded frequency is compared with its expected projection. In hypothesis testing, the Chi-square is calculated by comparing the observed frequencies in a dataset with the frequencies that would be expected if there were no significant associations between the variables. If the discrepancy between the observed and expected frequencies is substantial, it can be concluded that there is a significant association between the variables, thus leading to the rejection of the null hypothesis (H\({}_{0}\)). In this case, we could propose two hypotheses. The null hypothesis, H\({}_{0}\), suggests no statistical association between the observed valley streamflows, \(Q_{V}\), and valley streamflows for different return periods, \(Q_{V,Tt}\). Alternatively, H\({}_{1}\) establishes a statistical association between \(Q_{V}\) and \(Q_{VTt}\)[23; 26]. The coefficient of variation, \(C_{V}\)[12], measures the variability in a data sample in relation to its mean value. The skewness coefficient, \(C_{S}\)[12], assesses the symmetry of the data, measuring the third moment around the mean, indicating the direction and magnitude in which the tail of the distribution extends to the right or left relative to its mean value. If the tail is to the left, it is called negative skewness, and if it is to the right, it is called positive skewness. The kurtosis coefficient, \(C_{K}\)[27], is a statistical measure used to describe the shape of the probability distribution of a dataset. It indicates the data distribution around the mean and the comparison between the tails of the distribution and the shape of the normal distribution. A positive kurtosis coefficient, \(C_{K}\), is