Twitxy Saved

Peyman Milanfar

Verified account This account is verified. Learn more

@docmilanfar

Distinguished Scientist at Google. National Academy of Engineering. Computational Imaging ∩ AI. Posts are personal opinions
milanfar.org Joined February 2014
617 Following 119K Followers 8.8K Posts

Latest posts 5

Peyman Milanfar Verified account @docmilanfar
When x and y are independent random variables, their joint cumulative dist function (CDF) is the product of the individual CDFs:

F(x, y) - F(x)*F(y) = 0

Can uncorrelatedness (Cov(x,y) =0) also be expressed in terms of the CDFs? Yes.

This is due to the beautiful and little-known identity by Hoeffding (1940):

Cov(x,y) = ∫∫[F(x, y) - F(x)*F(y)] dxdy

It says that the difference between independence and uncorrelated-ness comes down to point-wise equality vs. a (weaker) integral equality between the CDFs.
3 replies 32 reposts 391 likes 24K views View on X