Let X and Y be jointly distributed random variables. This

ISBN: 9780073401331 38

Solution for problem 29E Chapter 2.6

Statistics for Engineers and Scientists | 4th Edition

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 -11.

Using these we have to prove the others.

Step 2 of 4

Step 3 of 4

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