Boundsforthecorrelation:

(a) Considerrandom variables X and Y and theirstandardizedvariables Zx and Zy. Using the equationsfromthepreviousexerciseandtherelationbetweenthecorrelationand covariance,showthatvar(Zx +Zy) ≥ 0 implies thatcorr(X,Y ) ≥ −1 and var(Zx −Zy) ≥ 0 implies28 corr(X,Y ) ≤ +1.

(b) Whenthejointdistributionissuchthatnecessarily X = Y , showthatcorr(X,Y ) = 1.