Consider an rv X with mean m and standard deviation s,

Chapter 4, Problem 124

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Consider an rv X with mean m and standard deviation s, andlet g(X) be a specified function of X. The first-order Taylorseries approximation to g(X) in the neighborhood of m isg(X) < g(m) 1 g9(m) ? (X 2 m)The right-hand side of this equation is a linear functionof X. If the distribution of X is concentrated in an intervalover which g(?) is approximately linear [e.g., x isapproximately linear in (1, 2)], then the equation yieldsapproximations to E(g(X)) and V(g(X)).a. Give expressions for these approximations. [Hint:Use rules of expected value and variance for a linearfunction aX 1 b.]b. If the voltage v across a medium is fixed but currentI is random, then resistance will also be a randomvariable related to I by R 5 v/I. If mI 5 20 andsI 5 .5, calculate approximations to mR and sR