The oxygen equivalence number of a weld is a number that can be used to predict properties such as hardness, strength, and ductility. The article “Advances in Oxygen Equivalence Equations for Predicting the Properties of Titanium Welds” (D. Harwig, W. Ittiwattana, and H. Castner, The Welding Journal, 2001:126s 136s) presents several equations for computing the oxygen equivalence number of a weld. An equation designed to predict the strength of a weld is X = 1.12C + 2.69N + O − 0.21 Fe, where X is the oxygen equivalence, and C, N, O, and Fe are the amounts of carbon, nitrogen, oxygen, and iron, respectively, in weight percent, in the weld. Suppose that for welds of a certain type, μ_C = 0.0247, μ_N = 0.0255, μ_O = 0.1668, μ_Fe = 0.0597, σ_C = 0.0131, σ_N = 0.0194, σ_O = 0.0340, and σ_Fe = 0.0413. Furthermore assume that correlations are given by ρ_C,N = −0.44, ρ_C,O = 0.58, ρ_C,Fe = 0.39, ρ_N,O = −0.32, ρ_N,Fe = 0.09, and ρ_O,Fe = −0.35.

a. Find μ_X.

b. Find Cov(C, N), Cov(C, O), Cov(C, Fe), Cov(N, O), Cov(N, Fe), and Cov(O, Fe).

c. Find σ_X.