1 minute read

The Intuition

For a Gaussian (Normal) pair of random variables, Pearson’s correlation coefficient ($\rho$) fully captures dependence. Step outside the Gaussian world, though, and $\rho$ can fail spectacularly.

Before diving in, recall the classical definition:

\[\rho_{X,Y}=\frac{\operatorname{cov}(X,Y)}{\sigma_X\sigma_Y},\]

where $\sigma$ denotes the standard deviation. Values are bound in the range $[-1, 1]$. A value of 0 is often mistaken for “independence” — but it only guarantees no linear relationship.

A Simple Counter-Example

Let’s consider two random variables:

  • $X \sim \mathcal{N}(0,1)$
  • $Y = \lvert X \rvert$

Intuitively, $Y$ is completely determined by $X$, so they are dependent. Let’s simulate this and see what Pearson’s $\rho$ says.

import numpy as np
import pandas as pd

n = 100_000
x = np.random.normal(0, 1, n)
df = pd.DataFrame({"X": x, "Y": np.abs(x)})

print(df.corr())

The expected output will be something like this (values will fluctuate slightly):

  X Y
X 1.000000 0.000???
Y 0.000??? 1.000000

As you can see, $\rho$ is approximately $0$! Pearson’s correlation fails here because the dependence is non-linear and symmetric around zero.

Takeaways

  • Zero correlation does not imply independence (unless your variables are jointly Gaussian).
  • Always visualize your data or use a non-linear measure of correlation (e.g., Spearman, Kendall, distance correlation, mutual information).
  • In risk analysis or alpha research, relying solely on Pearson’s correlation may hide important tail dependencies or regime shifts.

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