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Consider the discrete random variables $X,Y,Z, W$ respectively with supports $\{x_1, x_2\}$, $\{y_1, y_2\}$, $\{z_1, z_2, z_3\}$, $\{w_1, w_2, w_3\}$.

Assume $$ Pr(X=x_1)\leq Pr(Z=z_1)+Pr(Z=z_2) \hspace{1cm} [a] $$ $$ Pr(X=x_2)\leq Pr(Z=z_3)+Pr(Z=z_2) \hspace{1cm} [b] $$ $$ 1=Pr(X=x_1)+Pr(X=x_2)= Pr(Z=z_1)+Pr(Z=z_2)+Pr(Z=z_3) \hspace{1cm} [c] $$ $$ Pr(Y=y_1)\leq Pr(W=w_1)+Pr(W=w_2) \hspace{1cm} [d] $$ $$ Pr(Y=y_2)\leq Pr(W=w_3)+Pr(W=w_2) \hspace{1cm} [e] $$ $$ 1=Pr(Y=y_1)+Pr(Y=y_2)= Pr(W=w_1)+Pr(W=w_2)+Pr(W=w_3) \hspace{1cm} [f] $$

I want to show that the above inequalities imply that $$ (\star) \hspace{1cm} Pr(X=x_1)*Pr(Y=y_1)+Pr(X=x_2)*Pr(Y=y_2) $$ $$ \leq \Big[Pr(Z=z_1)+Pr(Z=z_2)\Big]*\Big[Pr(W=w_1)+Pr(W=w_2)\Big] $$ $$ +\Big[Pr(Z=z_3)+Pr(Z=z_2)\Big]*\Big[Pr(W=w_3)+Pr(W=w_2)\Big] $$ $$ -Pr(Z=z_2)*Pr(W=w_2) $$


My thoughts: by mixing [a], [b], [d], [e] it is easy to show that $$ Pr(X=x_1)*Pr(Y=y_1)+Pr(X=x_2)*Pr(Y=y_2) $$ $$ \leq \Big[Pr(Z=z_1)+Pr(Z=z_2)\Big]*\Big[Pr(W=w_1)+Pr(W=w_2)\Big] $$ $$ +\Big[Pr(Z=z_3)+Pr(Z=z_2)\Big]*\Big[Pr(W=w_3)+Pr(W=w_2)\Big] $$ Nevertheless, it seems to me impossible to show that ($\star$) holds. If you could provide a set of numbers satisfying [a], [b], [c], [d], [e], [f] but violating ($\star$), I would gratefully accept that as an answer.

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  • $\begingroup$ Any suggestion of a counterexample? $\endgroup$ – TEX Apr 19 '17 at 18:40
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For convenience, $x_1$ is used to denote $\Pr(X = x_1)$ and similarly for other probabilities. We have $$ (x_1 + x_2)(y_1 + y_2) = (w_1 + w_2 + w_3) (z_1 + z_2 + z_3) $$ from which we obtain $$ \small x_1y_1 + x_2y_2 + \color{red}{x_2y_1 + x_1y_2} = (w_1 + w_2)(z_1 + z_2) + (w_3 + w_2)(z_3 + z_2) - z_2w_2 + \color{red}{z_3w_1 + w_3z_1} \tag{$\spadesuit$} $$ Since $x_1 \leq z_1 + z_2$, therefore $$ x_2 = 1 - x_1 \geq 1 - z_1 - z_2 = z_3 $$ Similarly, $$ x_1 \geq z_1, \quad y_2 \geq w_3, \quad\text{and}\quad y_1 \geq w_1 $$ Thus $$ x_2y_1 + x_1y_2 \geq z_3w_1 + z_1w_3 $$ where the LHS and the RHS are those red in $(\spadesuit)$. We conclude that $$ x_1y_1 + x_2y_2 \leq (w_1 + w_2)(z_1 + z_2) + (w_3 + w_2)(z_3 + z_2) - z_2w_2 $$

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