## 4 Confidence Interval of Damage: Non-Stationary Case
### Confidence Interval on the Sum of Independent Normal Random Variables  
The following results follow the same reasoning used for the confidence interval on the difference in expected values [27; 30]. As a starting point, and with the purpose of illustrating the approach, it is useful to discuss first the simplest case of the sum \(X_{1}+X_{2}\) of two independent and normally distributed random variables \(X_{1}\) and \(X_{2}\) with expected values \(\mu_{1}\), \(\mu_{2}\) and variances \(\sigma_{1}^{2}\), \(\sigma_{2}^{2}\). The goal is to construct a confidence interval on \(\mu_{1}+\mu_{2}\) in the hypothesis of different and unknown variances \(\sigma_{1}^{2}\), \(\sigma_{2}^{2}\). Inference will be based on two random samples of size \(n_{1}\) and \(n_{2}\) from \(X_{1}\) and \(X_{2}\), denoted as \(X_{1p}\), \(p=1,\ldots,n_{1}\) and \(X_{2q}\), \(q=1,\ldots,n_{2}\). Under the previous hypotheses on \(X_{i}\)'s, it follows that \(Z=X_{1}+X_{2}\) is normally distributed with expected value \(\mu_{Z}=\mu_{1}+\mu_{2}\) and variance \(\sigma_{Z}^{2}=\sigma_{1}^{2}+\sigma_{2}^{2}\)[27]. A logical point estimator of \(\mu_{Z}\) is the sum of sample means \(\overline{X}_{1}+\overline{X}_{2}\), each one defined as \(\overline{X}_{i}=(1/n_{i})\sum\limits_{j=1}^{n_{i}}X_{ij}\). It is well known that each \(\overline{X}_{i}\) is normally distributed with expected value \(\mu_{i}\) and variance \(\sigma_{i}^{2}/n_{i}\)[27]. Since the sample means \(\overline{X}_{1}\) and \(\overline{X}_{2}\) of independent random variables are also independent, the estimator \(\overline{X}_{1}+\overline{X}_{2}\) has expected value \(E\left[\overline{X}_{1}+\overline{X}_{2}\right]=\mu_{1}+\mu_{2}\) and variance \(Var\left[\overline{X}_{1}+\overline{X}_{2}\right]=\sigma_{1}^{2}/n_{1}+\sigma _{2}^{2}/n_{2}\). In order to derive a confidence interval on \(\mu_{Z}=\mu_{1}+\mu_{2}\), and taking advantage of the above findings, it is useful to consider the statistic:  
\[T_{\eta}=\frac{\overline{X}_{1}+\overline{X}_{2}-(\mu_{1}+\mu_{2})}{\sqrt{ \frac{S_{1}^{2}}{n_{1}}+\frac{S_{1}^{2}}{n_{2}}}} \tag{10}\]which has _approximately_ a \(t\) distribution with \(\eta\) degrees of freedom [27; 30]:  
\[\eta\cong\frac{\left(\frac{s_{1}^{2}}{n_{1}}+\frac{s_{2}^{2}}{n_{2}}\right)^{2}}{ \frac{\left(s_{1}^{2}/n_{1}\right)^{2}}{n_{1}-1}+\frac{\left(s_{2}^{2}/n_{1} \right)^{2}}{n_{2}-1}} \tag{11}\]  
If not integer, \(\eta\) has to be rounded down to the nearest integer [27]. In previous expressions, \(S_{i}^{2}\) is the sample variance of the \(i\)-th sample \(X_{ij}\), where \((n_{1}-1)S_{i}^{2}=\sum_{j=1}^{n_{i}}\left(X_{ij}-\overline{X}_{i}\right)\). Note that symbol \(s_{i}^{2}\) in Equation (11) denotes the numerical value of \(S_{i}^{2}\) computed from an observed sample of values \(x_{i1},x_{i2},\ldots,x_{in_{i}}\); as usual in statistics, lowercase letters indicate observed values of random variables or estimators, indicated by uppercase letters. From the distribution of \(T_{\eta}\) and the probability statement \(P\Big{(}-t_{\alpha/2,\eta}\leq T_{\eta}\leq t_{\alpha,\eta}\Big{)}=1-\alpha\), it is possible to construct the approximate \(100(1-\alpha)\%\) confidence interval on the sum of expected values \(\mu_{1}+\mu_{2}\) as  
\[(\overline{x}_{1}+\overline{x}_{2})-t_{\alpha/2,\eta}\sqrt{\frac{s_{1}^{2}}{ n_{1}}+\frac{s_{2}^{2}}{n_{2}}}\ \leq\ \mu_{1}+\mu_{2}\ \leq\ (\overline{x}_{1}+\overline{x}_{2})+t_{\alpha/2,\eta}\sqrt{\frac{s_{1}^{2}}{ n_{1}}+\frac{s_{2}^{2}}{n_{2}}} \tag{12}\]  
where \(t_{\alpha/2,\eta}\) is the upper \(\alpha/2\) percentage point of the \(t\) distribution with \(\eta\) degrees of freedom given in Equation (11). Symbols \(\overline{x}_{1}\), \(\overline{x}_{2}\) and \(s_{1}^{2}\), \(s_{2}^{2}\) indicate the numerical values, respectively, of the sample mean and sample variance calculated from the two observed samples of \(X_{1}\) and \(X_{2}\) (by definition, the sample mean is \(\overline{x}_{i}=(1/n_{i})\sum_{j=1}^{n_{i}}x_{ij}\)).