Let $w=X*r$ with $X$ a real number and $w$ a random variable; let $u=mean(w)-var(w)$. What is the $X$ that gives the maximum $u$? What are the features of $r$ that are reasonable to ask in order to solve this problem?
How about if $w = \Sigma_i(X_i*r_i)$ where $i = 1,2,...,n$
We have $$u=\mathbb{E}(w)-Var(w) = X\mathbb{E}(r) - X^2Var(r),$$ this a quadratic function and the maximum of this function is obtained when $X=\frac{\mathbb{E}(r)}{2Var(r)}$.
Now let $w = \sum\limits_{i=1}^nX_i r_i$ where $r_i$ independent random variables, hence $$u = \sum\limits_{i=1}^nX_i \mathbb{E}(r_i) - \sum\limits_{i=1}^nX_i^2 Var(r_i)=\sum\limits_{i=1}^n\Big(X_i \mathbb{E}(r_i) - X_i^2 Var(r_i)\Big),$$ can you go from here?