Independent solution

How to solve this Covariance and Correlation question

Setup

Setup

Use the deterministic relationship to express the covariance through moments of the first variable.

Y=X2Y=X^2
Cov⁡(X,Y)=E[X3]−E[X]E[X2]\operatorname{Cov}(X,Y)=E[X^3]-E[X]E[X^2]

Model

Model

Evaluate the moments from the three equally weighted support points.

E[X]=−1+0+13=0E[X]=\frac{-1+0+1}{3}=0
E[X2]=1+0+13=23E[X^2]=\frac{1+0+1}{3}=\frac23
E[X3]=−1+0+13=0E[X^3]=\frac{-1+0+1}{3}=0

Compute

Compute

Substitute the moments, then test independence with a joint event rather than inferring it from covariance.

Cov⁡(X,Y)=0−0(23)=0\operatorname{Cov}(X,Y)=0-0\left(\frac23\right)=0
Pr⁡(X=0,Y=0)=13≠Pr⁡(X=0)Pr⁡(Y=0)=19\Pr(X=0,Y=0)=\frac13\ne\Pr(X=0)\Pr(Y=0)=\frac19

Answer

Answer

The variables are uncorrelated but dependent.

Cov⁡(X,Y)=0 and dependence(C)\boxed{\operatorname{Cov}(X,Y)=0\ \text{and dependence}\quad\text{(C)}}