FOSM Approximation of Mean

For the mean, we proceed by computing the expected value of the Taylor expansion:

$$\begin{array}{rl}\mathbb{E}[\mathcal{M}(\bar{X})+{\mathcal{M}}^{\prime }(\bar{X})\tilde{X}]& =\mathbb{E}[\mathcal{M}(\bar{X})]+\mathbb{E}[{\mathcal{M}}^{\prime }(\bar{X})\tilde{X}]\\ & =\mathcal{M}(\bar{X})+{\mathcal{M}}^{\prime }(\bar{X})\text{cancelto}0\mathbb{E}[\tilde{X}]\end{array}$$ $$\boxed{\mathbb{E}[\mathcal{M}(X)]\approx \mathcal{M}(\bar{X})}$$

This shows that evaluating the model at the mean input is an approximation of the mean output.