Exercise 2.15 (Schwarzs Inequality) For any random variables X and Y with finite variances 2 X

Exercise 2.15 (Schwarz’s Inequality) For any random variables X and Y with finite variances σ²X and σ²Y

, respectively, prove that

and that an equality holds only if Y = aX + b for some constants a ̸= 0 and

b. Because of this and (2.26), we have −1 ≤ ρ[X, Y ] ≤ 1 and ρ[X, Y ] = ±1 if and only if X = ±aY + b with a > 0.

Transcribed Image Text:

|C[X,Y] 0X0Y, ,