---
tag: 077GN9HDR6
kind: statement
role: proposition
name: Independent random variables are uncorrelated
msc: [60A05]
uses: [0DJ8W3HP2D, 05AW1G0WE1, 0E9502NT92]
sources:
  - url: https://en.wikipedia.org/wiki/Uncorrelatedness
    locator: Definition for two real random variables
  - url: https://en.wikipedia.org/wiki/Covariance
    locator: Uncorrelatedness and independence
---

## Statement

Let $X$ and $Y$ be [[independent|0E9502NT92]] real random variables with $E[X^2] < \infty$ and $E[Y^2] < \infty$.
Then $\operatorname{Cov}(X, Y) = 0$, so $X$ and $Y$ are uncorrelated.

## Remarks

Proof: by the [[product rule for independent variables|05AW1G0WE1]], $E[XY] = E[X] \, E[Y]$. So $\operatorname{Cov}(X, Y) = E[XY] - E[X] \, E[Y] = 0$.
The converse is false. The node [[uncorrelated random variables that are not independent|0C6J8BQXVR]] gives a counterexample.
For a pair $(X, Y)$ with a joint multivariate normal law, uncorrelated does imply independent.

## Explanations

### L1 human

If two variables do not influence each other at all, they have no linear relation either.
But "no linear relation" is a much weaker property than independence.