Problem 257 - An Introduction to Mathematical Statistics and Its Applications

Question

For continuous random variables \(X\) and \(Y\), prove that \(E(Y)=E_{X}[E(Y \mid x)]\).

Step-by-Step Solution

Answer

The proof is complete after showing that the right-hand side of the equation \(E_{X}[E(Y|X)]\) simplifies to \(E(Y)\), the left-hand side of the equation, thus demonstrating that \(E(Y) = E_{X}[E(Y|X)]\).

Step 1: Understanding the Problem Statement

First, understand that you are given two continuous random variables, \(X\) and \(Y\), and asked to prove that \(E(Y) = E_{X}[E(Y | X)]\). This equation is a form of the law of iterated expectations, which says that the expectation of a random variable (in this case, \(Y\)) equals the expected values of its conditional expectations.

Step 2: Using the Definition of Conditional Expectation

You apply the definition of conditional expectation to the random variable \(Y\) given \(X\). This is expressed as \(E(Y | X) = \int y f(y|x) dy\), where \(f(y|x)\) is the conditional probability density function of \(Y\) given \(X\).

Step 3: Using the Definition of Expectation

Then, you apply the definition of expectation, substituting in the definition of conditional expectation. This gives \(E_{X}[E(Y|X)] = \int (\int y f(y|x) dy) f(x) dx\), where \(f(x)\) is the probability density function of \(X\).

Step 4: Simplifying the Expression

Simplify the expression using the property of integration. The equation becomes \(E_{X}[E(Y|X)] = \int y (\int f(y|x) dx) dy\), which can be recognized as the joint density function of \(X\) and \(Y\). This simplifies to \(E_{X}[E(Y|X)] = \int y f(y) dy\), where \(f(y)\) is the probability density function of \(Y\).

Step 5: Concluding the Proof

This last equation simply equals \(E(Y)\), according to the definition of expectation, hence you can conclude that \(E(Y) = E_{X}[E(Y|X)]\). This completes your proof.