Consider an rv X with mean m and standard deviation s, and let g(X) be a specified function of X. The first-order Taylor series approximation to g(X) in the neighborhood of m is

g(X) ≈ g(μ) + gꞌ(m) ∙ (X - m)

The right-hand side of this equation is a linear function of X. If the distribution of X is concentrated in an interval over which g(∙) is approximately linear [e.g., √x is approximately linear in (1, 2)], then the equation yields approximations to E(g(X)) and V(g(X)).

a. Give expressions for these approximations. [Hint: Use rules of expected value and variance for a linear function aX + b.]

b. 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 = ν / I. If μI = 20 and σI = .5, calculate approximations to μR and σR.