To derive the conditional expectation $E(X \mid Y)$ using the given expressions and introduce the covariance term naturally, follow these steps:

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Step 1: Express Y in terms of X and Z

We have: $Y = X + Z,$ where $X \sim \mathcal{N}(0, \sigma_X^2)$, $Z \sim \mathcal{N}(0, \sigma_Z^2)$, and they are independent, so $\text{Cov}(X, Z) = 0$.

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Step 2: Formulate the conditional expectation $E(X \mid Y)$

Since $X$ and $Z$ are Gaussian random variables and $Y$ is a linear combination of them, the conditional expectation $E(X \mid Y)$ is linear in $Y$. Thus: $E(X \mid Y) = aY + b,$ where $a$ and $b$ are constants to be determined.

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Step 3: Compute $E(X \mid Y)$ using properties of joint distributions

1. The joint distribution of $(X, Y)$ is bivariate normal because $X$ and $Z$ are Gaussian.
2. The covariance matrix of $(X, Y)$ is: \sigma_X^2 & \sigma_X^2 \\ \sigma_X^2 & \sigma_X^2 + \sigma_Z^2 \end{pmatrix}.$$
3. The conditional expectation of $X$ given $Y$ for a bivariate normal distribution is: $E(X \mid Y) = \frac{\text{Cov}(X, Y)}{\text{Var}(Y)} Y = \frac{\sigma_X^2}{\sigma_X^2 + \sigma_Z^2} Y.$

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Step 4: Substitute the relation $X = Y - Z$

Since $X = Y - Z$, using this expression directly:

$E(X \mid Y) = E(Y - Z \mid Y) = Y - E(Z \mid Y).$

Because $Z$ is independent of $X$, the conditional expectation $E(Z \mid Y)$ depends on its correlation with $Y$. Specifically:

$E(Z \mid Y) = \frac{\sigma_Z^2}{\sigma_X^2 + \sigma_Z^2} Y.$

Substituting this into the expression for $E(X \mid Y)$:

$E(X \mid Y) = Y - \frac{\sigma_Z^2}{\sigma_X^2 + \sigma_Z^2} Y = \frac{\sigma_X^2}{\sigma_X^2 + \sigma_Z^2} Y.$

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Step 5: Covariance interpretation

The term $\text{Cov}(X, Y)$ appears in the formula for $E(X \mid Y)$: $\text{Cov}(X, Y) = \sigma_X^2,$ so: $E(X \mid Y) = \frac{\text{Cov}(X, Y)}{\text{Var}(Y)} Y,$ where: $\text{Var}(Y) = \sigma_X^2 + \sigma_Z^2.$

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Conclusion

By using $X = Y - Z$, the covariance term arises naturally in the conditional expectation. The expression:

$E(X \mid Y) = \frac{\sigma_X^2}{\sigma_X^2 + \sigma_Z^2} Y$ demonstrates the dependency of $X$ on $Y$ weighted by the relative variances (covariances) of the components.

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Do you have any questions or would you like further clarifications?
Here are five related questions and one tip for further exploration:

1. How can we generalize this method to non-Gaussian distributions?
2. Why is the conditional expectation linear for jointly Gaussian variables?
3. How does $E(Z \mid Y)$ simplify given the independence of $X$ and $Z$?
4. How can we derive $\text{Cov}(X, Y)$ explicitly if the variances are unknown?
5. What would happen if $Z$ had a different mean, say $\mu_Z \neq 0$?

Tip: Always verify independence assumptions when computing conditional expectations involving sums of random variables.