To prove that $\mathbb{E}(Y | X) = \mathbb{E}(Y)$ when $X$ and $Y$ are independent, we proceed as follows:

Definitions and Setup

• $\mathbb{E}(Y | X)$: The conditional expectation of $Y$ given $X$.
• Independence: $X$ and $Y$ are independent if $P(X, Y) = P(X)P(Y)$ for all values of $X$ and $Y$.

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Proof

1. Definition of Conditional Expectation: By definition: $\mathbb{E}(Y | X) = \int_{-\infty}^\infty y \, f_{Y|X}(y|x) \, dy,$ where $f_{Y|X}(y|x)$ is the conditional probability density function of $Y$ given $X$.

2. Independence Implies Factorization: When $X$ and $Y$ are independent, the joint probability density function factorizes as: $f_{X,Y}(x, y) = f_X(x)f_Y(y),$ and the conditional density becomes: $f_{Y|X}(y|x) = \frac{f_{X,Y}(x, y)}{f_X(x)} = \frac{f_X(x)f_Y(y)}{f_X(x)} = f_Y(y).$

3. Substitute $f_{Y|X}(y|x) = f_Y(y)$: Substituting into the formula for conditional expectation: $\mathbb{E}(Y | X) = \int_{-\infty}^\infty y \, f_{Y|X}(y|x) \, dy = \int_{-\infty}^\infty y \, f_Y(y) \, dy.$

4. Simplify the Integral: The integral $\int_{-\infty}^\infty y \, f_Y(y) \, dy$ is the definition of $\mathbb{E}(Y)$. Thus: $\mathbb{E}(Y | X) = \mathbb{E}(Y).$

5. Conclusion: Since the result does not depend on $X$, it follows that: $\mathbb{E}(Y | X) = \mathbb{E}(Y).$

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Summary

We have shown that if $X$ and $Y$ are independent, then $\mathbb{E}(Y | X) = \mathbb{E}(Y)$, as the conditional density $f_{Y|X}(y|x)$ reduces to the marginal density $f_Y(y)$, making the conditional expectation equal to the unconditional expectation.

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Do you have any questions or need further details on this proof?
Here are 5 related questions you might find useful:

1. What is the definition of conditional expectation and how does it generalize the idea of expectation?
2. How is independence formally defined in terms of probability density functions?
3. Can you provide an example where $\mathbb{E}(Y | X) \neq \mathbb{E}(Y)$ when $X$ and $Y$ are not independent?
4. How does the concept of conditional expectation relate to the law of total expectation?
5. What are some practical applications of conditional expectation in real-world scenarios?

Tip: Independence simplifies many statistical and probabilistic calculations by allowing separability of joint distributions.