Weighted sum of identical distributed random variables Suppose $X_1$, $X_2$, $\ldots$ ,$X_N$ are identically distributed (not necessarily independent). 
Then, given $a_1+a_2+\ldots+a_N=1$, and let $S=a_1 X_1 + a_2 X_2 + \ldots + a_N X_N$. Does $S$ follow the same distribution as $X$?

Let the $X_i$ be independent and identically distributed, with non-zero variance $\sigma^2$.  Suppose that the $a_i$ are $\ge 0$, and at least $2$ of them are non-zero. Then $a_1X_1+a_2X_2+\cdots a_NX_N$ has variance $(a_1^2+\cdots+a_N^2)\sigma^2$, which is less than $\sigma^2$. 
