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Suppose that we have integer random variables $X>0$ and $Y>0$ and constant number $a$. We have: $X+Y < a$. Can we say that the covariance of these random variables is less than or equal to zero? Because they are negatively correlated?

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You can't. Here's an example: suppose X is uniformly distributed between 0 and 1. $Cov(X, X)$ is strictly positive, but we still have $X + X < a$ if $a$ is anything greater than 2.

This example is with a continuous variable, you specified you had discrete variables (integers) but this is the same idea.

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To give a more general answer:

Suppose that you have two postitive and positively correlated, bounded Random Variables $X,Y$ with $X<C_1,Y<C_2$. Now assume that $a> 0$.

You can simply set: $\tilde{X}:=\frac{X}{C_1}\cdot a/2,\quad \tilde{Y}:=\frac{Y}{C_2}\cdot a/2$

Now you know that $cor(\tilde{X},\tilde{Y})>0$ (as Correlation is invariant under affine linear transformations) and therefore $cov(\tilde{X},\tilde{Y})>0$. Furthermore:

$\tilde{X}+\tilde{Y}<\frac{X}{C_1}\cdot a/2+\tilde{Y}:=\frac{Y}{C_2}\cdot a/2<a$, as $X<C_1,Y<C_2$

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