Relative difference between joint probability and product of marginals.

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Let $p_i,j;igeq 1, jgeq 1$ be a joint probability distribution, and $p_i,cdot;igeq 1$ and $p_cdot,j;jgeq 1$ be two corresponding marginal distributions. Does the following inequality always hold?
$$ left(sum_i,jp_i,j^2-sum_ip_i,cdot^2 times sum_jp_cdot,j^2right)^2leq
left(sum_i,jp_i,j^2-sum_i,jp_i,j^2 times sum_i,jp_i,j^2right)^2$$



Note that the troublesome issue in proving this inequality (if it is true) is that the expression within the parenthesis on the left, $sum_i,jp_i,j^2-sum_ip_i,cdot^2 times sum_jp_cdot,j^2$, could be positive (in which case the proof is easy) or negative (in which case I don't see a proof).










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    Let $p_i,j;igeq 1, jgeq 1$ be a joint probability distribution, and $p_i,cdot;igeq 1$ and $p_cdot,j;jgeq 1$ be two corresponding marginal distributions. Does the following inequality always hold?
    $$ left(sum_i,jp_i,j^2-sum_ip_i,cdot^2 times sum_jp_cdot,j^2right)^2leq
    left(sum_i,jp_i,j^2-sum_i,jp_i,j^2 times sum_i,jp_i,j^2right)^2$$



    Note that the troublesome issue in proving this inequality (if it is true) is that the expression within the parenthesis on the left, $sum_i,jp_i,j^2-sum_ip_i,cdot^2 times sum_jp_cdot,j^2$, could be positive (in which case the proof is easy) or negative (in which case I don't see a proof).










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      Let $p_i,j;igeq 1, jgeq 1$ be a joint probability distribution, and $p_i,cdot;igeq 1$ and $p_cdot,j;jgeq 1$ be two corresponding marginal distributions. Does the following inequality always hold?
      $$ left(sum_i,jp_i,j^2-sum_ip_i,cdot^2 times sum_jp_cdot,j^2right)^2leq
      left(sum_i,jp_i,j^2-sum_i,jp_i,j^2 times sum_i,jp_i,j^2right)^2$$



      Note that the troublesome issue in proving this inequality (if it is true) is that the expression within the parenthesis on the left, $sum_i,jp_i,j^2-sum_ip_i,cdot^2 times sum_jp_cdot,j^2$, could be positive (in which case the proof is easy) or negative (in which case I don't see a proof).










      share|cite|improve this question















      Let $p_i,j;igeq 1, jgeq 1$ be a joint probability distribution, and $p_i,cdot;igeq 1$ and $p_cdot,j;jgeq 1$ be two corresponding marginal distributions. Does the following inequality always hold?
      $$ left(sum_i,jp_i,j^2-sum_ip_i,cdot^2 times sum_jp_cdot,j^2right)^2leq
      left(sum_i,jp_i,j^2-sum_i,jp_i,j^2 times sum_i,jp_i,j^2right)^2$$



      Note that the troublesome issue in proving this inequality (if it is true) is that the expression within the parenthesis on the left, $sum_i,jp_i,j^2-sum_ip_i,cdot^2 times sum_jp_cdot,j^2$, could be positive (in which case the proof is easy) or negative (in which case I don't see a proof).







      probability functional-analysis






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      edited Sep 7 at 10:53

























      asked Sep 7 at 10:40









      Student of Statistics

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