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Chapter 7: Q.27 (page 361)

Prove that if E[Y鈭�X=x]=E[Y]for all x, then Xand Yare uncorrelated; give a counterexample to show that the converse is not true.

Hint: Prove and use the fact that E[XY]=E[XE[Y鈭�X]].

Short Answer

Expert verified

We prove that,E[Y鈭�X=x]=E[Y]for allx

Step by step solution

01

Given information

Given in the question that, we have to prove thatE[Y鈭�X=x]=E[Y]for allx

02

Explanation

Let's find the covariance between Xand Y. We have that

Cov鈦�(X,Y)=E(XY)鈭�E(X)E(Y)

Since we have that E(Y鈭�X=x)=E(Y)for all xes, we have that

E(Y鈭�X)=E(Y)

Which implies

Hence

Cov鈦�(X,Y)=0

03

Disapprove the converse

Now, let's disapprove the converse. Consider random variable X~N(0,1)and define Y=X2. We have that

Cov鈦�(X,Y)=Cov鈦�X,X2=EX3鈭�E(X)EX2

Observe that X3is symmetric around zero, so EX3=0. Finally, we see that Cov(X,Y)=0. But, we have that

E(Y鈭�X)=EX2鈭�X=X2

which is not obviously equal to constant E(Y)=EX2.

04

Final answer

We prove that, E[Y鈭�X=x]=E[Y]for all x

And from the above example we prove thatE(Y)=EX2

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