Let $f(x,y)$ be a joint density of $X$ and $Y$. Let $f_{X}(x)$ and $f_{Y}(y)$ be the marginal density functions. Is the following inequality true? $$ \int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}f^2(x,y)dxdy\geq \int_{-\infty}^{+\infty}f^2_{X}(x)dx\times \int_{-\infty}^{+\infty}f^2_{Y}(y)dy. $$
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1$\begingroup$ Could you say a bit about the motivation? $\endgroup$– Aryeh KontorovichCommented Sep 1, 2017 at 10:19
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1$\begingroup$ For discrete distributions this seems not to work try (0.7 0.15) and (0.15, 0) as the two rows of a bivariate distribution where X and Y take two values. $\endgroup$– martin crippsCommented Sep 1, 2017 at 13:19
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$\begingroup$ Martin Cripps, you are right! That is indeed a counter-example. The claim is not true!!! Thanks. $\endgroup$– Student of StatisticsCommented Sep 1, 2017 at 13:40
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$\begingroup$ Regardless of the motivation, it appears that something like the claimed inequality might hold with some universal constant $c$ (just not 1, as the counterexample indicates). $\endgroup$– Aryeh KontorovichCommented Sep 2, 2017 at 20:26
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