Let X and Y be jointly distributed random variables. This

QUESTION:

Let X and Y be jointly distributed random variables. This exercise leads you through a proof of the fact that \(-1 \leq \rho_{X, Y} \leq 1\).

a. Express the quantity \(V\left(X-\left(\sigma_{X} / \sigma_{Y}\right) Y\right)\) in terms of \(\sigma_{X}\), \(\sigma_{Y}\), and \(\operatorname{Cov}(X, Y)\).

b. Use the fact that \(V\left(X-\left(\sigma_{X} / \sigma_{Y}\right) Y\right) \geq 0\) and \(\operatorname{Cov}(X, Y)=\rho_{X, Y} \sigma_{X} \sigma_{Y}\) to show that \(\rho_{X, Y} \leq 1\).

c. Repeat parts (a) and (b) using \(V\left(X+\left(\sigma_{X} / \sigma_{Y}\right) Y\right)\) to show that \(\rho_{X, Y} \geq-1\).

Equation Transcription:

Text Transcription:

-1{</=}rho_{X,Y}{</=}1

V(X-(sigma_X/sigma_Y)Y)

sigma_X

sigma_Y

Cov(X,Y)

V(X-(sigma_X/sigma_Y)Y){>/=}0

Cov(X,=rho_{X,Y}sigma_Xsigma_Y

rho_{X,Y}{</=}1

V(X+(sigma_X/sigma_Y)Y)

rho_{X,Y}{>/=}-1