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

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.