## 3 Materials and Methods
### Data Driven Analysis Using Probability Distribution Model  
Once we get \(\phi\), then we will move to the third step. In this step, data-driven analysis is performed for computing:  
1. Lower and upper bound for each group of each category and
2. Out of pocket employees before and after optimization
3. Estimation of total medical benefits amount by using a probability distribution model  
A probability distribution is a function that depicts the possibility of getting the values that a random variable can have within the distribution. In our case, this random variable is the \(amount\). We evaluate the results of the above two steps, with the help of probability distribution models. The probability distribution model that fits employee insurance records is the Gamma distribution model as shown in Figure 4. On the X-axis, there number of employees and on the Y-axis is the probability density function. Figure 4: Probability distribution model for employees insurance records. The shape and scale parameters of gamma distributions are a and b respectively. The formula for Mean \(\mu\) and standard deviation \(\sigma\) in terms of gamma distribution parameters are represented in Equations (4) and (5). \[\mu_{G_{h}}=a_{G_{h}}b_{G_{h}} \tag{4}\]  
\[\sigma_{G_{h}}=\frac{\sqrt{a_{G_{h}}}}{b_{G_{h}}} \tag{5}\]  
where h = {1,2,3,4..}, depending upon the number of groups within the category. Once we get \(\mu_{G_{h}}\) and \(\sigma_{G_{h}}\) for each group of all categories, we can adjust the premium amount for each group of every category, by using a gamma distribution plot. Algorithm 3 explains the computation of lower and upper bound for premium amounts for each category using the gamma probability distribution model. \(V^{i}\) is the out of pocket employees where i represents a category. ```
Input: \(\{\phi_{1},\phi_{2}...\_7\}\) for(\(i\gets 1\) to 7)do \(\mu^{i}=a^{i}b^{i}\) \(\sigma^{i}=\frac{\sqrt{a^{i}}}{b^{i}}\) for(\(h\gets 1\) to \(G\)do \(\mu^{i}_{G_{h}}=a^{i}_{G_{h}}b^{i}_{G_{h}}\) \(\sigma^{i}_{G_{h}}=\frac{\sqrt{a^{i}_{G_{h}}}}{b^{i}_{G_{h}}}\)  end for for(\(f\gets 1\) to 3)do if(\(T<\phi\))then \(\mu^{i}+f*\sigma^{i}\)  end if  end for Set lower bound= \(\mu^{i}\)  Set Upper bound= \(\mu^{i}+1*\sigma^{i}\) for(\(s\gets 2\) to 3)do \(V^{i}\)+=[P(\(\mu^{i}+s*\sigma^{i}\))*(\(\mu^{i}+s*\sigma^{i}\))*100  end for  end for
```  
**Algorithm 3**Data driven analysis using probability distribution Model. The main topic of discussion nowadays is that most employers or organizations are paying the excess premium amount to the insurance companies while their employees are not frequently consuming the amount of the medical benefit. The objective is to align the premium amount with actual medical utilization made by each category of employees. The historical medical records enable us to generate need-based packages, with the help of these packages we can estimate actual utilization of amount using the probability distribution concept. After data visualization we observed that the gamma distribution fits the data as shown in Figure 4. The process of optimization depends on the difference between \(T\) real medical benefits amount and \(\phi\) computed medical benefits amount. The data-driven analysis using a probability distribution plot is performed for each group in the specified category. For each group, we evaluate the mean \(\mu\) and standard deviation \(\sigma\). It is observed that as the amount increases the number of employees availing that amount is reduced. This fact can be observed from the shape of the gamma distribution curve. When we get a value for a standard deviation for each group, we know this fact that standard deviation is used as a constant in the probability distribution. Within the gamma distribution, we can move towards the right side of the mean and can get estimates of how many employeesare getting out of pocket while availing healthcare services. All medical expenses equal and greater than \(\mu+2\sigma\) are not affordable for employees.