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I have a form like X(X+Y-1) for X and Y independent random variables. I believe I can show $E[X+Y-1]>0$ and $E[X]>0$. Am I allowed to conclude that also $E[X(X+Y-1)]>0$? I have heard a fkg inequality, but don't know if it can be used in this multiple variables case Thank you in advance for any hints!

toki
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$$E[X(X+Y-1)] = E[X^2] + E[XY] -E[X] \geq E[X]^2 +E[X]E[Y] -E[X] = E[X](E[X] + E[Y] - 1) = E[X] E[X+Y-1] > 0 $$

paperskilltrees
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  • Thank you very much, but this inequality does not hold in general? – toki Dec 23 '23 at 04:02
  • I am not sure what situation you refer to when you say "in general". Here we used independence and non-negativity of variance. – paperskilltrees Dec 23 '23 at 04:06
  • I mean $E[XY]\geq E[X]E[Y]$ if both $E[X]$ and $E[Y]$ positive, even if we don't know much about X and Y? – toki Dec 23 '23 at 04:08
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    Note that $E[XY] - E[X]E[Y] = \mathrm{cov}[X,Y]$, which is limited only by variances of $X, Y$, unless further conditions are imposed. – paperskilltrees Dec 23 '23 at 04:16