property of the joint probability of two discrete random variables

Question

I have two discrete random variables :

$X$ with probabilities $p_i$ with $1 \le i \le N_X$

$Y$ with probabilities $q_j$ with $1 \le j \le N_Y$

let $r_{ij}$ stands for the joint probability $Pr(X=i,Y=j)$

I want to compare the following :

$A=\sum_{ij}(r_{ij}^2)$

$B=\sum_{i}(p_{i}^2).\sum_{j}(q_{j}^2)$

From many tests with $N_X=N_Y=3$, I have only got $A \ge B$. But, I could not devise a proof.

I tried to use convexity of $f(x,y)=xy$ or $g(x)=x^2$

Answer 1

It is provable using the convexity of $f(x,y)=xy$.

just write $r_{ij}^2=p_i h(j/i) q_j h(i/j)$ where h are conditional probabilities.

Then use Jensen's inequality with weights $p_iq_j$ (sums to 1)

$A \ge \sum_{ij}(p_iq_jh(i/j))\sum_{ij}(p_iq_jh(j/i))=B$