Sum of normally distibuted random variables? I feel terribly confused over this.
If:
X~N(μx,σx^2)
Y~N(μy,σy^2)
both independent random variables
Z=X+Y
Z~N(μx+μy,σx^2+σy^2)
Then why:
Xi~N(μi,σi^2)
i=1, 2, ..., n
X1, X2, ..., Xn are independent random variables
Y=c1X1+c2X2+...+cnXn
E(Y)=c1μ1+c2μ2+...+cnμn
V=(c1^2)σ1^2+(c2^2)σ2^2+...+(cn^2)σn^2
In the first case, imagine if X=Y:
Z=X+X
Z=2X
Z~N(2μx,2(σx^2))
But by the second one:
Z=2X
Z~(2μx,2^2*(σx^2))
Why... Is the variance in both cases different?
Or are both properties completely unrelated and I'm doing something dumb?

The variable $X$ is not independent of variable $X$. Thus the correlation coefficient is unequal to zero. 
The variance of the sum of two dependent, normally distributed random variables (X,Y) is 
$\sigma_{X+Y}^2=\sigma_X^2+\sigma_Y^2+2\rho \sigma_X\cdot \sigma_Y$
Setting Y equal to X.
$\sigma_{X+X}^2=\sigma_X^2+\sigma_X^2+2\rho_X \cdot \sigma_X\cdot \sigma_X$
Note, that $\rho_X=Corr(X,X)=1$
$\sigma_{X+X}^2=\sigma_X^2+\sigma_X^2+2\cdot 1 \cdot \sigma_X\cdot \sigma_X=4\sigma_X^2$
