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Variance of a Linear Combination of Variables

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Let X, Y, and Z be random variables with the following means and standard deviations: μX=10,σX=2\mu_X = 10, \sigma_X = 2; μY=15,σY=3\mu_Y = 15, \sigma_Y = 3; and μZ=20,σZ=4\mu_Z = 20, \sigma_Z = 4. The correlation between any pair of these variables is ρ=0.5\rho = 0.5. What is the value of Var(X+Y−2Z)\mathrm{Var}(X + Y - 2Z)?

A

4 + 9 + 16 = 29

B

4 + 9 + 64 + 6 - 16 - 24 = 43

C

4 + 9 + 16(4) = 77

D

(2)² + (3)² + (-2)²(4)² = 73

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