Variance of $X-Y$ cannot be negative

Question

Suppose we have 2 non independent variable X and Y.

Since $$Var(X-Y)= Var(X) + Var (Y) -2Cov(X,Y)$$

Would it be possible for the above to be negative in the case when $$Var(X) + Var(Y) < 2 Cov(X,Y)$$?

Answer 1

You could use the fact that a variance is non-negative, i.e. $$var(X - Y) \geq 0$$ So $$var(X) + var(Y) - 2cov(X,Y) \geq 0$$ Hence $$var(X) + var(Y) \geq 2cov(X,Y) $$

Answer 2

$\newcommand{\v}{\operatorname{var}}$No. You have $$ \v(X) + \v(Y) - 2\operatorname{cov}(X,Y) = \v(X-Y) \ge 0 $$ and therefore $$ \operatorname{cov}(X,Y) \le \frac {\v(X) + \v(Y)} 2. $$ If the sum of the variances is less than $2,$ then the covariance is less than $1.$