Past exam of the mathematics course of the University of Cambridge / 2010 / ia / Paper 2 / 4F / b

Zero correlation means that the variables are uncorrelated: $E[(X-EX)(Y-EY)]=0$, or equivalently $E(XY)=EXEY$. It does not imply independence. For example, if $X$ is uniform on $[-1,1]$ and $Y=X^2$, symmetry gives $EX=E(X^3)=0$, hence zero covariance, although $Y$ is determined by $X$. Both random variables are non-constant and have finite positive variances.

Perfect correlation means an affine relation almost surely: there are constants $a\ne 0,b$ such that $Y=aX+b$ almost surely. The sign of $a$ is the sign of the correlation coefficient. More precisely, equality in the Cauchy-Schwarz inequality gives