## 3 Methodology
### Estimate of the Best-Fit Probability Distribution  
Estimating the best-fit probability distributions in reliability involves selecting a distribution that closely matches the observed data. It is important to consider the characteristics of the data and the assumptions of different distributions when selecting the most appropriate method for estimating the best-fit probability distribution in reliability analysis. There are several methods available for this purpose. The best-fit probability distribution is generally selected based on the goodness-of-fit results obtained from the Kolmogorov-Smirnov (K-S) test [41], Least Square Regression (LSR) [42], and Maximum Likelihood Estimation (MLE) [43]. In this study, the Weibull++ (version 2023) tool developed by HBK Prenscia (Duluth, GA, USA) is used for the analysis of TTF data. The tool performs five \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Probability Distribution** & **Parameters** & **Reliability Function** & \multicolumn{2}{c}{**Characteristics of the**} \\  & & & & **Distribution** \\ \hline \multirow{4}{*}{\begin{tabular}{c} Weibull \\ 3-parameter \\ \end{tabular} } & \(\beta\)—shape parameter & & \(\bullet\) & Mostly used for reliability analysis. \\  & \(\gamma\)—location parameter or assured & & \(\bullet\) & This distribution has the ability to take the shape of another distribution. \\  & life & \(\theta\)—scale parameter & & \(\bullet\) & It can be used to model decreasing, constant, and increasing hazard rates. \\ \hline \multirow{4}{*}{\begin{tabular}{c} [PERSON] \\ 2-parameter \\ \end{tabular} } & \(\beta\)—shape parameter & \(R(t)=e^{-(\frac{t}{k})^{\beta}}\) & \(\bullet\) & The location parameter is zero. \\  & \(\theta\)—scale parameter & & \(R(t)=e^{-\lambda(t-\gamma)}\) & \(\bullet\) & The hazard rate is constant. \\  & \(\gamma\)—assured life & & \(\bullet\) & The failures are random. \\ \hline \multirow{4}{*}{\begin{tabular}{c} Exponential \\ 1-parameter \\ \end{tabular} } & \(\lambda\)—failure rate & \(R(t)=e^{-\lambda t}\) & \(\bullet\) & The assured life is equal to zero. \\ \hline \multirow{4}{*}{\begin{tabular}{c} Lognormal \\ \(t_{\text{\tiny{med}}}\) \\ \end{tabular} } & \(s\)—shape parameter or standard & \multirow{4}{*}{\(R(t)=1-\Phi\left(\frac{1}{k}ln\frac{t}{t_{\text{\tiny{med}}}}\right)\)} & \(\bullet\) & Its logarithm is normally distributed. \\  & deviation & & The Probability Density Function (PDF) takes \\  & \(t_{\text{\tiny{med}}}\)—location parameter & & & Various shapes, which is just like the Weibull distribution. \\ \hline \multirow{4}{*}{\
\begin{tabular}{c} Normal (Gaussian) \\ \end{tabular} } & \(\mu\)—mean or & \multirow{4}{*}{\(R(t)=1-\phi\left(\frac{t-p}{\phi}\right)\)} & \(\bullet\) & Widely used in statistics and data analysis. \\  & Mean-Time-To-Failure (MTTF) & & \(\bullet\) & A large number of natural phenomena follow the normal distribution. \\  & parameter & & & \\ \hline \multirow{4}{*}{
\begin{tabular}{c} Gamma \\ \end{tabular} } & \(\eta\)—shape parameter & \multirow{4}{*}{\(R(t)=\sum\limits_{k=0}^{n-1}\frac{(\lambda t)^{k}e^{(-\lambda t)}}{k!}\)} & \(\bullet\) & It reduces to exponential distribution for \(\eta=1\). \\  & \(\lambda\)—scale parameter & & \(\bullet\) & It is often used to estimate redundant system reliability with identical components having a constant failure rate. \\  & & & If \(\eta\) is an integer, then it is called an Erlang distribution. \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Probability distributions and their key characteristics. goodness-of-fit (GOF) tests (K-S test, correlation coefficient, likelihood value, Bayesian information criterion, and Akaike information criterion) to decide the best-fit probability distribution. This approach is robust and accurately validates the reliability estimates and offers practical implications, failure prediction, and system optimization, especially in the case of an HVAC system.