1

In this post I would like someone to summarize and relate these 3 concepts of statistics (in the context of stats).

1) I remember that uncorrelated does NOT imply independence (e.g. the case where the RV $X=Y^2$, correlation=0 but they are dependent in a nonlinear way. (as mentioned here:https://en.wikipedia.org/wiki/Uncorrelated_random_variables)

2)However can we say that independent implies uncorrelated, i.e. Pearson corr. coef. = 0 ?

3)For orthogonality: - Does orthogonal imply uncorrelated ? (i think yes) - Does uncorrelated imply orthogonal?

4) There are definitions of orthogonal w.r.t. Expected value as: $E[XY]$=0

A definition of uncorrelated is: $E[XY]=E[X]E[Y]$

how can we relate both (e.g. with an example) ?

5) Finally, what to say if X and Y are vector RVs ? Does $E[XY]$ goes down to a dot product of X and Y ? ( I m trying to refresh an reorganize my knowledge in my head so would be great to have like a "summary post" or "cheat sheet" here)

SheppLogan
  • 152
  • 9
  • A sec of googling yields https://stats.stackexchange.com/questions/171324/what-is-the-relationship-between-orthogonal-correlation-and-independence – Mark L. Stone Apr 01 '18 at 12:06
  • thanks yes I know that post, which is good, but I am not entirely satisfied an wanted to create another one here. It doen't hurt to summarize briefly again those concepts i think – SheppLogan Apr 01 '18 at 12:08
  • the idea here would be to briefly (e.g. ~1 sentence) summarize each for example – SheppLogan Apr 01 '18 at 12:08
  • Moreover, after reading posts like : https://stats.stackexchange.com/questions/12128/what-does-orthogonal-mean-in-the-context-of-statistics it seems especially that the concept of orthogonality in statistics is still often confusing, i.e., what does it really mean for 2 RV to be "at right angle"? – SheppLogan Apr 01 '18 at 13:30
  • according to A. Donda in the post you mentioned, Orthogonality in the context of statistics would be the same as correlation: $ \langle X, Y \rangle = \mathrm{cov} (X, Y) = E [ (X - E[X]) (Y - E[Y]) ].$ However, i would like more discussion on that ... – SheppLogan Apr 01 '18 at 13:34
  • note for myself: https://www.quora.com/What-is-orthogonality-in-statistics – SheppLogan Apr 01 '18 at 13:41
  • 2
    Can you explain what you do not understand in that linked post? Just rewriting the same here that you do not understand there do no good. – kjetil b halvorsen Apr 01 '18 at 15:24
  • The idea was to develop the concept further, especially w.r.t. geometrical interpretation of orthognonality. But the other post is good, no doubts there, I think i will stay with the definition of 2 RV orthogonal if $\langle X, Y \rangle = \mathrm{cov} (X, Y) = E [ (X - E[X]) (Y - E[Y]) ] = 0.$ guess – SheppLogan Apr 05 '18 at 13:02

0 Answers0