To calculate the expectation \( E[P[X | (X+Y)]] \), let's start by analyzing the given notation. 

Let:
- \( X \) and \( Y \) be two random variables.
- We are interested in finding \( E[P[X | (X+Y)]] \), which suggests the expectation of the probability of \( X \) given the sum \( X + Y \).

### Step 1: Understanding \( P[X | (X+Y)] \)
If \( P[X | (X+Y)] \) represents the conditional probability of \( X \) given \( X+Y \), then we need the joint distribution of \( X \) and \( Y \) to proceed, or at least some assumptions about their relationship (e.g., independence, specific distribution type).

### Step 2: Setting Up the Expectation
The expectation \( E[P[X | (X+Y)]] \) can be computed as:
\[
E[P[X | (X+Y)]] = \int_{-\infty}^{\infty} P[X | (X+Y) = s] f_{X+Y}(s) \, ds
\]
where \( f_{X+Y}(s) \) is the probability density function of \( X+Y \).

### Step 3: Special Cases
1. **If \( X \) and \( Y \) are independent and normally distributed**:
   - If \( X \sim \mathcal{N}(\mu_X, \sigma_X^2) \) and \( Y \sim \mathcal{N}(\mu_Y, \sigma_Y^2) \), then \( X+Y \sim \mathcal{N}(\mu_X + \mu_Y, \sigma_X^2 + \sigma_Y^2) \).
   - The conditional expectation can be derived using properties of normal distributions.

2. **If \( X \) and \( Y \) follow other specific distributions (e.g., exponential or uniform)**:
   - The approach will differ according to the properties of these distributions.

If you can provide more details on the distributions or assumptions on \( X \) and \( Y \), I can proceed with a more specific calculation. 

Would you like further details on these calculations or assumptions for particular distributions? 

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**Additional Related Questions:**
1. How do we calculate conditional probabilities for dependent variables?
2. What is the expectation of a function of random variables?
3. How does independence affect the conditional expectation calculation?
4. Can we calculate \( E[P[X | (X+Y)]] \) if \( X \) and \( Y \) are jointly uniform?
5. What is the role of moment-generating functions in expectation calculations?

**Tip:** When dealing with conditional probabilities and expectations, knowing whether random variables are independent or not simplifies calculations significantly.

Explore the calculation of the expectation of conditional probability P[X | (X+Y)]. This solution provides a detailed breakdown of key probability concepts, including conditional expectation, joint distribution, and assumptions for common distributions like normal. Ideal for college-level probability and statistics students.

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