Director, Senior Director, even VP titles have lost their currency in the AI world. They’ve become recruiting tools and retention carrots - Silicon Valley’s own version of grade inflation.
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.