## 2 Derivation of Reliability Evaluation Theory  
The stress-strength interference model is often used to evaluate the reliability of pyrotechnic devices [PERSON] et al. (2018). Factors that cause products' failure are regarded as stress, while factors that prevent products' failure are regarded as strength. Usually the strength of the products is higher than the stress, but the stress and strength have characteristics of random dispersion, which produces an \"interference zone\" in their distribution curves as shown in the shaded area in Figure 1. In this area, unreliable situations happen, which result in the function failure [PERSON] et al. (2018); [PERSON] et al. (2018); [PERSON] et al. (2018). Sometimes, when the products have undergone a strict process of quality control, the products' stress has high repeatability and small discreteness, and it can be treated as a constant, which results in a simplified stress-strength model, as shown in Figure 2.  
In Figure 2, \(L\) represents the stress in the stress-strength model, which is a constant, \(f\left(x\right)\) is the probability density function of the random variable \(X\), and \(X\) is assumed to be distributed normally. Assuming that \(X_{1}\), \(X_{2}\),..., \(X_{n}\) are samples from the normal population \(X\)\(\sim\)\(N(\mu,\,\sigma^{2})\), which are independent basic random variables. \(\overline{X}\) and \(S\) are the sample mean and sample standard deviation, respectively, and the lower specification limit of the product is \(L\). Introducing the \(K\) factor:  
\[K=\left(\overline{X}-L\right)/S \tag{1}\]  
Figure 1: The stress–strength interference model. After taking n observations of the random variable \(X\), we are interested in the observation value \(n\) + 1 and later, but no observation is made at \(n+1\) or it is temporarily unable to be observed; therefore, we can predict it based on the previous n observations. \(Y_{1}\), \(Y_{2}\),..., \(Y_{k}\) are the future \(k\) independent observations of \(X\) and \(\overline{Y}\) is the average of \(k\) samples. Then, there is \(\overline{Y}=\sum\limits_{j=1}^{k}Y_{j}/k\)  
According to the knowledge of mathematical statistics, it can be known that the sample mean of the random variable \(X\) obeys the normal distribution. Then, there are \(\overline{X}\sim N(\mu,\sigma^{2}/n),\overline{Y}\sim N(\mu,\sigma^{2}/k)\). Introduce the random variable \(Z=\overline{X}-\overline{Y}\). The mean and variance of \(Z\) can be obtained as:  
\[E(Z)=E(\overline{X}-\overline{Y})=E(\overline{X})-E(\overline{Y})=0,\;D(Z)=D( \overline{X}-\overline{Y})=D(\overline{X})+D(\overline{Y})=\sigma^{2}(\frac{1} {n}+\frac{1}{k})\]  
Then, there is \(Z\sim N(0,\frac{\sigma^{2}}{n}+\frac{\sigma^{2}}{k})\). Transforming the random variable \(Z\) into a standard normal distribution:  
\[Z^{\prime}=\frac{Z-E(Z)}{\sqrt{D(Z)}}=\frac{\overline{X}-\overline{Y}}{\sigma \sqrt{(1/n)+(1/k)}}\sim N(0,\;1)\]  
From the knowledge of mathematical statistics:  
\[\frac{(n-1)S^{2}}{\sigma^{2}}\sim\chi^{2}(n-1) \tag{2}\]  
Therefore, according to the definition of \(t\) distribution:  
Assuming that \(A\) and \(B\) are independent random variables, \(A\sim N(0,1),B\sim\chi^{2}(n)\), then the random variable \(m=A/\sqrt{B/n}\) obeys the \(t\) distribution with n degrees of freedom. It is denoted as \(m\)-\(t(n)\). Let random variable \(T=\frac{Z^{\prime}}{S/\sigma}\), considering \(Z^{\prime}\) as \(A\) and \((n-1)S^{2}/\sigma^{2}\) as \(B\), we can get:  
\[T=\frac{Z^{\prime}}{S/\sigma}=\frac{\overline{X}-\overline{Y}}{S\sqrt{(1/n)+( 1/k)}}\sim t(n-1) \tag{3}\]  
That is, the random variable \(T\) obeys the \(t\) distribution with \(n-1\) degrees of freedom. Since the pyrotechnic device has a one-shot function, it only needs to work once when performing a task, so let \(k=1\). Its reliability is the probability that the next observed value in the future is greater than the lower specification limit, namely:  
\[R=\Pr(\overline{Y}>L) \tag{4}\]  
Figure 2: Simplified model of the stress–strength interference. where, Pr represents the probability.