FOSM

Approximation of Mean

• A random variable in terms of deterministic and stochastic components is given by

$$X=\bar{X}+\tilde{X}$$

Here we assume that $E[X]=\bar X$ and $E[\tilde X]=0$

• Using first order Taylor Polynomial

$$\mathcal{M}(X)=\mathcal{M}(\bar{X}+\tilde{X})\approx \mathcal{M}(\bar{X})+{\mathcal{M}}^{\prime }(\bar{X})\tilde{X}$$

• Expectation

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

Approximation of Moment

$$\boxed{\mathbb{V}[\mathcal{M}(X)]\approx {\left({\mathcal{M}}^{\prime }(\bar{X})\right)}^2\mathbb{V}[\tilde{X}]}$$

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