On: Academics Make Theoretical Breakthrough in Random ...

[wfunction]

Isn't "Independent and no correlations" redundant? How can two random variables be independent but correlated?

[cjslep → wfunction]

Dependence is stricter than correlation: the population violates probabilistic independence. Correlation can help guide statisticians to finding a dependence between variables, because correlation measures how close or how far a sample is to independence. But this gives rise to the "Correlation does not imply causation" adage.

Example of independent but correlated variables: http://www.tylervigen.com/spurious-correlations

[stan_rogers → wfunction]

An easy example is two thermal sources in the same environment.

[prashnts → wfunction]

Not really. Zero correlation does not necessarily imply independence. From the example on this resource[1]:

Let X be a normally distributed random variable with zero mean, and say Y = X^2. Clearly they are not independent. Covariance, which is needed for (pearson) correlation coefficient, can be calculated to be 0:

    Cov(X, Y) = E(XY) - E(X)E(Y)

              = E(X^3) - 0        (Since E(X) = mean(X) = 0)

              = 0                 (Since X is centered at 0)

[1] http://mathforum.org/library/drmath/view/64808.html

[gizmo686 → wfunction]

The paper itself does not explicitly require there to be "no correlations". My guess is that that phrase was added by the journalist as emphasis.