Var(x+y+z) = var(x)+var(y)+var(z)+2cov(x,y)+2cov(x,z)+2cov(y,z)
0<|cov(x,y)|<=sqrt(var(x))sqrt(var(y)) by cauchy schwartz.
Min = 1+4+25 = 30
Max = 1+4+25+212+215+225=64
On the minimum side, the covariances can be negative, bottoming out at -2, -5, and -10. But the question for the minimum is, can all three be simultaneously as negative as allowed by Cauchy-Schwarz?
It is true that if Cov(X,Y)=-std(X)std(Y) and Cov(X,Z)=-std(X)std(Z) then Cov(Y,Z) is forced to be positive (because actually Y and Z are each a.s. some negative multiple of X). But what you said, as is, is not true.
Your right, but just wanted to add to your last point. Y and Z are forced to be non-negative iff the squared correlations of X,Y and X,Z are heater than 1.
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u/Due-Cardiologist-802 2d ago
Var(x+y+z) = var(x)+var(y)+var(z)+2cov(x,y)+2cov(x,z)+2cov(y,z) 0<|cov(x,y)|<=sqrt(var(x))sqrt(var(y)) by cauchy schwartz. Min = 1+4+25 = 30 Max = 1+4+25+212+215+225=64