$\newcommand{\Var}{\operatorname{Var}}$ $\newcommand{\Cov}{\operatorname{Cov}}$ I wonder how come that the variance $\Var(sX)=s^2\Var(X)$, but $\Var\left(\sum_{i=1}^s X_i\right)=\sum_{i=1}^s \Var(X_i)+\sum_{i=1,j=1 ,i\neq j}^{s}2\Cov(X_i,X_j)=\sum_{i=1}^s \Var(X_i)=s\Var(X)$ for $s$ uncorrelated random variables with equal distribution. What is the main reason for that?
Thank you.