Relative difference between joint probability and product of marginals.
Clash Royale CLAN TAG#URR8PPP
up vote
2
down vote
favorite
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
add a comment |Â
up vote
2
down vote
favorite
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
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
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
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
probability functional-analysis
edited Sep 7 at 10:53
asked Sep 7 at 10:40
Student of Statistics
1719
1719
add a comment |Â
add a comment |Â
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2908498%2frelative-difference-between-joint-probability-and-product-of-marginals%23new-answer', 'question_page');
);
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password