Past exam of the mathematics course of the University of Cambridge / 2010 / ia / Paper 2 / 4F (Probability)

Put $U=X-EX$ and $V=Y-EY$. Their second moments are the finite positive variances of $X,Y$. The Cauchy-Schwarz inequality for square-integrable random variables givesDivide by the positive product of variances and take square roots. The correlation bound is $\boxed{-1\le \rho (X,Y)\le 1}$. As usual, non-constant here means not almost surely constant, so that the denominator is nonzero.

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

Interpret the random choice as an independent selector $B$ with $P(B=1)=r$, and write $Y=BX+(1-B)X^{\prime }$. Its distribution is again uniform on $\{-1,1\}$, so $EX=EY=0$ and $\mathrm{Var}X=\mathrm{Var}Y=1$. By independence,Consequently $\boxed{\rho (X,Y)=r}$. The independent selector is the standard meaning of the stated mixture; a choice depending on the signs would describe a different model.