Chapter 7: Q.27 (page 361)

Prove that if $E [Y ∣ X = x] = E [Y]$ for all $x$ , then $X$ and $Y$ are uncorrelated; give a counterexample to show that the converse is not true.
Hint: Prove and use the fact that $E [X Y] = E [X E [Y ∣ X]]$ .

Short Answer

Expert verified

We prove that, $E [Y ∣ X = x] = E [Y]$ for all $x$

Step by step solution

Given information

Given in the question that, we have to prove that $E [Y ∣ X = x] = E [Y]$ for all $x$

Explanation

Let's find the covariance between $X$ and $Y$ . We have that

$Cov (X, Y) = E (X Y) - E (X) E (Y)$

Since we have that $E (Y ∣ X = x) = E (Y)$ for all $x$ es, we have that

$E (Y ∣ X) = E (Y)$

Which implies

Hence

$Cov (X, Y) = 0$

Disapprove the converse

Now, let's disapprove the converse. Consider random variable $X ~ N (0, 1)$ and define $Y = X^{2}$ . We have that

$Cov (X, Y) = Cov (X, X^{2}) = E (X^{3}) - E (X) E (X^{2})$

Observe that $X^{3}$ is symmetric around zero, so $E (X^{3}) = 0$ . Finally, we see that $C o v (X, Y) = 0$ . But, we have that

$E (Y ∣ X) = E (X^{2} ∣ X) = X^{2}$

which is not obviously equal to constant $E (Y) = E (X^{2})$ .

Final answer

We prove that, $E [Y ∣ X = x] = E [Y]$ for all $x$

And from the above example we prove that $E (Y) = E (X^{2})$

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Prove that if $E [Y ∣ X = x] = E [Y]$ for all $x$ , then $X$ and $Y$ are uncorrelated; give a counterexample to show that the converse is not true.
Hint: Prove and use the fact that $E [X Y] = E [X E [Y ∣ X]]$ .

Short Answer

Step by step solution

Given information

Explanation

Disapprove the converse

Final answer

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Prove that if E[Y∣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∣X]].

Short Answer

Step by step solution

Given information

Explanation

Disapprove the converse

Final answer

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Applied Mathematics

Geometry

Theoretical and Mathematical Physics

Discrete Mathematics

Decision Maths

Study anywhere. Anytime. Across all devices.

Prove that if $E [Y ∣ X = x] = E [Y]$ for all $x$ , then $X$ and $Y$ are uncorrelated; give a counterexample to show that the converse is not true.
Hint: Prove and use the fact that $E [X Y] = E [X E [Y ∣ X]]$ .