Chaimaa Boukabouy retweetledi
We usually repeat that correlation doesn’t imply causation, which is true. But the reverse, causation doesn’t imply correlation, is ALSO true.
Let X~N(0,1) and Y=X^2. In this example, X causes Y.
But the correlation between X and Y is Corr(X,Y)=Corr(X,X^2)=Cov(X,X^2)/(σ_X*σ_(X^2))=(E[X*X^2]-E[X]*E[X^2])/(σ_X*σ_(X^2))=0. Why?👇
The function h(x)=x^3 is odd:
(-x)^3=-x^3
and the distribution is symmetric around 0, f is even:
f(x)=f(-x)
Thus (x^3)*f(x) is odd (odd*even=odd function) and the integral of an odd function over a symmetric interval is 0.
Thus E[X^3]=∫X^3*f(x)dx=0
(the integral is from -♾️ to ♾️)
The same occurs with the function g(x)=x because g(-x)=-x=-g(x) (it’s odd).
So E[X]=∫X*f(x)dx=0
(the integral is again from -♾️ to ♾️)
Thus Corr(X,Y)=0 because Cov(X,X^2)=0.
Thus we have proven that causation DOES NOT IMPLY correlation.
Q. E. D. ▪️
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