Template:Stress-Strength Confidence Intervals

Confidence Interval of the Probability

Both the stress and strenght distributions can be either estimated from actual data or specified by engineers based on engineering knowledge or existing references. Based on the sources of the distribution, there are two types of variations associated with the calculated probability.

Variation in Model Parameters

If both stress and strength distributions are estimated from data sets, then there are uncertainty associated with the estimated distribution parameters. These uncertainty will cause some degree of variation of the calculated probability from the stress-strength analysis. Therefore, we can use these uncertainty to estimate the confidence interval of the calculated probability. To get the confidence interval, we first calcualte the variance of the Reliablility using:

[math]\displaystyle{ Var\left[ R \right]=\int_{0}^{\infty }{Var\left[ {{f}_{1}}(x) \right]}{{\left[ {{R}_{2}}(x) \right]}^{2}}dx+{{\int_{0}^{\infty }{\left[ {{f}_{1}}(x) \right]}}^{2}}Var\left[ {{R}_{2}}(x) \right]dx }[/math]

Variance of [math]\displaystyle{ {{f}_{1}}(x) }[/math] and [math]\displaystyle{ {{R}_{2}}(x) }[/math] can be estimated from the Fisher Information Matrix. For detail, please see chapter Confidence Bounds.

Once the variance of the expected reliability is obtained, the two-sided confidence interval of it can be calcualted using:

[math]\displaystyle{ [\frac{R}{R+(1-R)w},\frac{R}{R+(1-R)/w}] }[/math]

where:

CL is the confidence level;

[math]\displaystyle{ \alpha }[/math] = 1-CL;

[math]\displaystyle{ w=\exp \{{{z}_{1-\alpha /2}}\sqrt{Var(R)}/[R(1-R)]\} }[/math];

[math]\displaystyle{ {{Z}_{1-\alpha /2}} }[/math] is the [math]\displaystyle{ 1-\alpha/2 }[/math] percential of a standard normal distribution.

If the upper bound (U) and lower bound (L) are not infinite and 0, the above calcualted variance of R is adjusted by [math]\displaystyle{ {{\left[ {1}/{\left( {{F}_{1}}(U)-{{F}_{1}}(L) \right)}\; \right]}^{2}} }[/math].

Variation in Probability Values

Assume the distributions for stress and strength are known. From the stress-strength equation:

[math]\displaystyle{ R=P[Stress\le Strength]=\int_{0}^{\infty }{{{f}_{Stress}}(x)\cdot {{R}_{Strength}}(x)}dx }[/math]

we know, the calcualted reliability is the Expected value of the probability that a strength value is larger than a stress value. Since both strength and stress are random variables from their distributions, the reliability is also a random variable. This can be explained using the following exampe. Let's first assume stress is a fixed value of 567. The reliability then is:

[math]\displaystyle{ R(567)=\Pr (Strength\gt 567)={{R}_{2}}(567) }[/math]

This is the reliability when the stress value is 567 and when the strength distribution is given. If stress is not a fixed value, instead it follows a distribution, then it can take values other than 567. For instance, it can be a value of 700. Therefore, we get another reliability value of [math]\displaystyle{ {R}(700) }[/math]. Since stress is a random variable, for any stress value [math]\displaystyle{ {x}_{i} }[/math], there is a reliability value of [math]\displaystyle{ R({{x}_{i}}) }[/math] calculated from the strength distribution. We will end up with many [math]\displaystyle{ R({{x}_{i}}) }[/math]s or [math]\displaystyle{ R_{2}({{x}_{i}}) }[/math]s. From these [math]\displaystyle{ R({{x}_{i}}) }[/math]s, we can get the mean and variance of the reliability. In fact, its mean is the the result from:

[math]\displaystyle{ R=P[Stress\le Strength]=\int_{0}^{\infty }{{{f}_{Stress}}(x)\cdot {{R}_{Strength}}(x)}dx }[/math]

and its variacne is:

[math]\displaystyle{ \begin{align} & Var\left[ R \right]=Var\left[ {{R}_{2}}({{X}_{1}}) \right]=E\left[ {{R}_{2}}{{\left( {{X}_{1}} \right)}^{2}} \right]-{{\left( E\left[ {{R}_{2}}\left( {{X}_{1}} \right) \right] \right)}^{2}} \\ & =\int_{0}^{\infty }{{{f}_{1}}(x){{\left[ {{R}_{2}}(x) \right]}^{2}}dx}-{{\left( E\left[ {{R}_{2}}\left( {{X}_{1}} \right) \right] \right)}^{2}} \\ & =\int_{0}^{\infty }{{{f}_{1}}(x){{\left[ {{R}_{2}}(x) \right]}^{2}}dx}-{{\left( R \right)}^{2}} \\ \end{align} }[/math]

where R is the expected value of the reliability.