FOSM Approximation of Variance

For the variance, we proceed by computing the variance of the Taylor expansion:

$$\begin{array}{rl}V[\mathcal{M}(X)]& \approx V\left[\mathcal{M}(\bar{X})+{\mathcal{M}}^{\prime }(\bar{X})\tilde{X}\right]\\ & =V\left[{\mathcal{M}}^{\prime }(\bar{X})\tilde{X}\right]\\ & ={\left({\mathcal{M}}^{\prime }(\bar{X})\right)}^{2}V[\tilde{X}]\end{array}$$ $$\boxed{V[\mathcal{M}(X)]\approx {\left({\mathcal{M}}^{\prime }(\bar{X})\right)}^{2}V[\tilde{X}]}$$

Where $\mathbb{E}[\widetilde{X^2}]=\mathbb{V}[\tilde{X}]$

This result shows that the output variance is approximately the product of the squared sensitivity (derivative) at the mean and the input variance, providing a simple way to propagate uncertainty through models.