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Probability with Applications in Engineering, Science, and Technology
The delta method provides approximations to the mean and variance of a nonlinear function $h(X)$ ) of a rv $X .$ These approximations are based on a first-order Taylor series expansion of $h(x)$ about $x=\mu,$ the mean of $X :$
$$h(X) \approx h_{1}(X)=h(\mu)+h^{\prime}(\mu)(X-\mu)$$
(a) Show that $E\left[h_{1}(X)\right]=h(\mu)$ . (This is the delta method approximation to $E[h(X)]$ .)
(b) Show that $\operatorname{Var}\left[h_{1}(X)\right]=\left[h^{\prime}(\mu)\right]^{2} \operatorname{Var}(X) .$ (This is the delta method approximation to $\operatorname{Var}[h(X)] . )$
(c) If the voltage $v$ across a medium is fixed but current $I$ is random, then resistance will also be a random variable related to $I$ by $R=v / I .$ If $\mu_{I}=20$ and $\sigma_{I}=.5,$ calculate approximations to $\mu_{R}$ and $\sigma_{R} .$
(d) Let $R$ have the distribution in Exercise $25,$ whose mean and variance are 10 and 1$/ 5$ respectively. Let $h(R)=\pi R^{2},$ the area of the ecologist's sampling region. How does $E[h(R)]$ from Exercise 25 compare to the delta method approximation $h(10) ?$
(e) It can be shown that $\operatorname{Var}[h(R)]=14008 \pi^{2} / 175 .$ Compute the delta method approximation to $\operatorname{Var}[h(R)]$ using the formula in $(\mathrm{b}) .$ How good is the approximation?