# Let X and Y be jointly distributed random variables. This ## Problem 29E Chapter 2.6

Statistics for Engineers and Scientists | 4th Edition

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Problem 29E

Let X and Y be jointly distributed random variables. This exercise leads you through a proof of the fact that -1 ≤ ρX,Y ≤ 1.

a. Express the quantity V(X − (σX/σY)Y) in terms of σX, σY, and Cov(X, Y).

b. Use the fact that V(X − (σX/σY)Y) ≥ 0and Cov(X, Y) = ρX,YσXσY to show that ρX,Y ≤ 1.

c. Repeat parts (a) and (b) using V(X + (σX/σY)Y) to show that ρX,Y ≥ −1.

Step-by-Step Solution:

Step 1 of 4:

It is given that X and Y are two random variables and are jointly distributed.

Also it is known that -1   1.

Using these we have to prove the others.

Step 2 of 4:

(a)

Here we have to express V(X-( )Y) in terms of , ,Cov(X,Y).

We know that

V(X)=E( )- Thus,

V(X-( )Y)=E - =E( -2 XY)- =E( )+E( )-E(2 XY)- + -2E(X)E( )]

=[E( )- ]-2 [E(XY)-E(x)E(Y)]+ [E( )- ]

= -2 Cov(X,Y)+  =2[ - Cov(X,Y)]

Hence,

V(X-( )Y)=2[ - Cov(X,Y)]

Step 3 of 4

Step 4 of 4

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