Minimize variance of $aX + bY + cZ$ for independent variables $X,Y,Z$ There are three independent random variables $X, Y, Z$.
The goal is to minimize the variance of $aX + bY + cZ$ where $a+b+c = 1$ and $0\le a,b,c \le1$.
$$\operatorname{var}(X) = 2\operatorname{var}(Y) = 3\operatorname{var}(Z)$$
I did $$K = aX + bY + cZ\quad,\quad \operatorname{var}(K) = a^2\operatorname{var}(X) + b^2\operatorname{var}(Y) + c^2\operatorname{var}(Z)\,,$$ but it looks like one more equation might be needed to solve the problem.
Is there anything I missed?

To minimize $6a^{2}+3b^{2}+2c^{2}$ subject to $a,b,c \geq 0$ and $a+b+c=1$ note that $1^{2}=(a+b+c)^{2}=(\frac 1 {\sqrt 6}\sqrt 6 a +\frac 1 {\sqrt 3}\sqrt 3 a +\frac 1 {\sqrt 2}\sqrt 2 a)^{2}  \leq (\frac 1 6 +\frac 1 3+\frac 1 2) (6a^{2}+3b^{2}+2c^{2})$ by C-S inequality. Hence $6a^{2}+3b^{2}+2c^{2} \geq 1$ and equality holds when $(a,b,c) =(\frac 1 6, \frac 1 3,\frac 1 2)$. Hence the minimum value is $1$. 
