## 4 Confidence Interval of Damage: Non-Stationary Case
### Confidence Interval on the Sum of Independent Normal Random Variables
It is now straightforward to generalize the result in Equation (12) to the case of a random variable \(Z=\sum_{i=1}^{N}X_{i}\) sum of \(N\) independent and normally distributed random variables \(X_{1},X_{2},\ldots,X_{N}\) with expected values \(\mu_{1},\mu_{2},\ldots,\mu_{N}\) and unequal variances \(\sigma_{1}^{2},\sigma_{2}^{2},\ldots,\sigma_{N}^{2}\). In this case, the approximate \(100(1-\alpha)\%\) confidence interval on the sum \(\mu_{Z}=\sum_{i=1}^{N}\mu_{i}\) is  
\[\left(\sum_{i=1}^{N}\overline{x}_{i}\right)-t_{\alpha/2,\
u}\sqrt{\sum_{i=1}^ {N}\frac{s_{i}^{2}}{n_{i}}}\ \leq\sum_{i=1}^{N}\mu_{i}\leq\ \left(\sum_{i=1}^{N}\overline{x}_{i}\right)+t_{ \alpha/2,\
u}\sqrt{\sum_{i=1}^{N}\frac{s_{i}^{2}}{n_{i}}} \tag{13}\]  
where \(\overline{x}_{i}\) and \(s_{i}^{2}\) denote the sample mean and sample variance of the \(i\)-th random sample \(x_{ij}\), \(j=1,\ldots,n_{i}\) with \(n_{i}\) observations for the random variable \(X_{i}\); \(t_{\alpha/2,\
u}\) is the upper \(\alpha/2\) percentage point of the \(t\) distribution with the following number of degrees of freedom:  
\[\
u\cong\frac{\left(\sum_{i=1}^{N}\frac{s_{i}^{2}}{n_{i}}\right)^{2}}{\sum_{i =1}^{N}\frac{\left(\frac{s_{i}^{2}}{n_{i}}\right)^{2}}{n_{i}-1}} \tag{14}\]  
As before, a fractional value of \(\
u\) must be rounded down to the nearest integer [27]. In the case, analyzed in Section 4.2, of equal sample sizes \(n_{i}=n\) (for any \(i\)), the previous expression simplifies as \(\
u\cong(n-1)\left(\sum_{i=1}^{N}s_{i}^{2}\right)^{2}/\sum_{i=1}^{N}\left(s_{ i}^{2}\right)^{2}\). The level of accuracy of the approximate confidence interval in Equation (13) was checked by numerical simulations in which various sets with different numbers of random variables \(X_{i}\), each with different expected value \(\mu_{i}\) and variance \(\sigma_{i}^{2}\), were examined. The selected values of \(\mu_{i}=0,\ 1,\ 2,\ 3\) and \(\sigma_{i}^{2}=1,\ 5,\ 10,\ 100\) allowed for a total of sixteen possible combinations \(\left(\mu_{i},\sigma_{i}^{2}\right)\) for the random variables \(X_{i}\). The first six are listed in Table 1.  
Not all random variables \(X_{i}\)'s were considered simultaneously; rather, their number was made to vary as \(N=2,\ 3,\dots,\ 16\), i.e., from only two variables up to all sixteen variables included. Once \(N\) has been selected, a random sample of size \(n\) was generated from the normal distribution of each \(X_{i}\) with parameters \(\mu_{i}\) and \(\sigma_{i}^{2}\). Different sample sizes in the range \(n=2,\ 3,\dots,\ 100\) were considered. For each combination of \(N\) and \(n\), the random sampling yielded \(N\) samples of size \(n\), the values of which were used to construct the 95% confidence interval as per Equation (13). This confidence interval, by definition, may or may not enclose the expected value \(\mu_{Z}=\sum_{i=1}^{N}\mu_{i}\). If the procedure is repeatedly applied, it is expected that 95% of confidence intervals will enclose \(\mu_{Z}\). To verify this assertion, the random sampling and construction of the confidence interval (for given combinations of \(N\) and \(n\)) were repeated \(2\cdot 10^{7}\) times; in this large set, the fraction of confidence intervals enclosing \(\mu_{Z}\) represented the estimated confidence level \(100(1-\hat{\alpha})\%\) to be compared with the theoretical value 95%. The outcome for a few selected combinations of \(\left(\mu_{i},\sigma_{i}^{2}\right)\), which are in fact well representative of a general trend, is summarized in Table 2. It is observed that, for \(N\) and \(n\) being both very small, Equation (13) gives confidence levels larger than the theoretical value 95%; for greater values of \(N\) and \(n\), the difference almost disappears. Only for the lowest values of \(n\), the difference may occasionally be large, for example, reaching 98.59% when \(n=2\) and \(N_{S}=16\). Nevertheless, this smallest sample size is only included here for comparative purposes; it being usually not used in practice.