Random variables satisfying a strict inequality

Solution 1:

1. In general, $\text{Var}(X+Y)=\text{Var}(X)+\text{Var}(Y)+2\text{Cov}(X,Y),$ so

$$\text{Var}(X+Y)\gtreqqless \text{Var}(X)+\text{Var}(Y)\iff \text{Cov}(X,Y)\gtreqqless 0.$$

So to ensure $\text{Var}(X+Y)> \text{Var}(X)+\text{Var}(Y)$, just choose $X$ and $Y$ positively correlated; $Y=X$ for nondegenerate $X$ is the easiest example that comes to mind.

2. As for standard deviations (where $\sigma(Z)\equiv +\sqrt{\text{Var}(Z)}$), note that $$\sigma(X+Y)\gtreqqless\sigma(X)+\sigma(Y)\iff \text{Var}(X+Y)\gtreqqless \text{Var}(X)+\text{Var}(Y)+2\sigma(X)\sigma(Y),$$

so by the above argument, it suffices to choose $Y$ and $X$ negatively correlated to ensure $\sigma(X+Y)<\sigma(X)+\sigma(Y)$; $Y=-X$ for nondegenerate $X$ is the easiest example that comes to mind.