Consider an rv X with mean and standard deviation μ, and let g(X) be a specified functio...

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

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.]

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 = v/I. If μ_I = 20 and σ₁ = .5 , calculate approximations to μ_R and σ_R .