FOSM Example

FOSM (First order second moment) for a measurement problem. Many quantities can only be measured indirectly. In these cases a measurement model

$$Z=g\left(X_1,X_2,\dots ,X_M\right)$$

is used, which connects the observable quantities $\mathbf{X}={\left(X_1,X_2,\dots ,X_M\right)}^{\top }$ with the not directly observable quantity $Z$. Erroneous measurements of $\mathbf{X}$ then lead to deviations in $Z$.

Consider the example of an electrical resistor whose resistance depends on the temperature. We denote with $V,\bar{R},\alpha ,T,\bar{T}$ the voltage, the resistance at $\bar{T}$, the linear coefficient of temperature, the current temperature and the reference temperature. For the measurement model there holds

$$P=g(V,\alpha ,T)=\frac{V^2}{\bar{R}(1+\alpha (T-\bar{T}))}$$

where $P$ denotes power. Use FOSM to find an approximation of the standard deviation of $P$, i.e. ${\sigma }_P$, in dependence of the known uncertainties of the directly observable quantities

$$\begin{array}{rl}& V\sim \mathcal{N}\left({\mu }_V,{\sigma }_V\right)\\ & \\ & \alpha \sim \mathcal{U}\left(\bar{\alpha }-{\Delta }_{\alpha },\bar{\alpha }+{\Delta }_{\alpha }\right)\\ & T\sim \mathcal{U}\left(\bar{T}-{\Delta }_T,\bar{T}+{\Delta }_T\right)\\ & \end{array}$$

All parameters of the given distributions as well as $\bar{R}$ can be considered

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