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.