Exercise · ZW-COVARIANCE-009 · ★★★ Advanced
Let \(X\) be uniform on \(\{-1,0,1\}\) and \(Y=X^2\). Which statement is correct?
Choose one option.
Show answer
- B. \(\operatorname{Cov}(X,Y)=0\) even though \(Y\) depends on \(X\)
Explanation
\(E[X]=0\) and \(E[XY]=E[X^3]=0\), so the covariance is 0; yet \(Y\) is determined by \(X\). Zero covariance does not imply independence.
Review the concept: Covariance
The covariance \(\operatorname{Cov}(X,Y)\) of two variables \(X\) and \(Y\) measures whether they tend to deviate from their expectations \(E\) in the same or opposite directions: it is the expectation of the product of their deviations.
\[\operatorname{Cov}(X,Y)=E\big[(X-E[X])(Y-E[Y])\big]\]
More practice problems: covariance
- ZW-COVARIANCE-001 E[X]=2,\ E[Y]=3 and E[XY]=7. Compute Cov(X,Y).
- ZW-COVARIANCE-002 If Var(X)=4, what is Cov(X,X)?
- ZW-COVARIANCE-003 Let X be uniform on \0,1,2\ and Y=2X. Compute Cov(X,Y).
- ZW-COVARIANCE-004 If X and Y are independent with E[X]=2 and E[Y]=5, what is Cov(X,Y)?
- ZW-COVARIANCE-005 If Cov(X,Y)=3, compute Cov(2X+1, -3Y+5).
- ZW-COVARIANCE-006 With the joint table P(0,0)=0.4,\ P(0,1)=0.1,\ P(1,0)=0.1,\ P(1,1)=0.4, compute Cov(X,Y).
- ZW-COVARIANCE-007 If Var(X)=4, Var(Y)=9 and Cov(X,Y)=-2, compute Var(X+Y).
- ZW-COVARIANCE-008 If Cov(X,Y)<0, what can be said?