I have been working on the following problem:

Given you have VarX = 1, VarY = 4, and VarZ = 25, what is the minimum possible variance for the random variable W = X + Y + Z, or min Var(X+Y+Z)?

My first thought is to complete the variance-covariance expansion as follows:

Var(X + Y + Z) = VarX + VarY + VarZ +2[Cov(X,Y) + Cov(Y,Z) + Cov(X,Z)]

Then to use the Cauchy-Schwarz inequality to determine the minimum covariance for each of the covariance terms (i.e. |Cov(X,Y)| <= sqrt(VarXVarY) ). However, I am obtaining a negative potential minimum, which leads me to think that the lower bound could be zero?

Var(X+Y+Z) = 1 + 4 + 25 + 2[-2 - 5 - 10] = 30 - 34 ???

The other thought is that using Cauchy-Schwarz in this way is not correct and my approach is wrong.

My next thought is to consider the expansion as Var[(X+Y), Z], but was not sure how to proceed by considering the sum of 2 variables (X+Y) and Z.

Any thoughts on how to proceed are appreciated.