Friday, September 20, 2013

On Zero Correlation and Statistical Independence

I put the following material together yesterday in response to a request from one of our grad. students. I thought it might be helpful to some readers of the blog.

© 2013, David E. Giles


  1. It is also can be seen from the independence=>zero correlation result, that independence can be equivalently written as X and Y are independent if E[g(X)f(Y)] = E[g(X)]E[f(Y)] for each pair of bounded and continuous functions g,f. It is clear then why zero correlation, or E[XY]=E[X]E[Y] is not sufficient for independence.

  2. A uniformly-distributed point on a chessboard is a nice example for this and some related things.

    - colour, row, and column are pairwise independent, but row and column completely determine colour.

    - colour is uncorrelated with, but perfectly determined by, diagonal position (ie, row+column)