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The formula given is a representation of the **conditional expectation** of a random variable \( Y \) given an event \( A \), where \( 1_A \) is the indicator function of the event \( A \). The formula is:

\[
E(Y \mid 1_A) = \frac{E(Y \cdot 1_A)}{P(A)}
\]

We will now prove this.

### Step-by-Step Proof:

1. **Definition of Conditional Expectation:**

   The conditional expectation \( E(Y \mid A) \) is defined as the expected value of \( Y \) given that the event \( A \) has occurred. Mathematically, this can be written as:

   \[
   E(Y \mid A) = \frac{E(Y \cdot 1_A)}{P(A)}
   \]

   where \( 1_A \) is the indicator function, which is 1 if the event \( A \) occurs and 0 otherwise.

2. **Using the Indicator Function:**

   The indicator function \( 1_A \) takes the value 1 when the event \( A \) occurs and 0 otherwise. Therefore, \( Y \cdot 1_A \) represents the value of \( Y \) only when \( A \) occurs and is 0 when \( A \) does not occur.

   \[
   Y \cdot 1_A = 
   \begin{cases}
   Y & \text{if } A \text{ occurs}, \\
   0 & \text{if } A \text{ does not occur}.
   \end{cases}
   \]

3. **Expectation of \( Y \cdot 1_A \):**

   The expectation \( E(Y \cdot 1_A) \) is the expected value of \( Y \) over the event \( A \), which can be written as:

   \[
   E(Y \cdot 1_A) = \int_A Y \, dP = \int_{\Omega} Y \cdot 1_A \, dP.
   \]

   This integral calculates the expected value of \( Y \) over the set where \( A \) happens.

4. **Normalization by \( P(A) \):**

   To compute the conditional expectation \( E(Y \mid A) \), we need to normalize the above expectation by the probability of the event \( A \), which ensures that we are conditioning on the occurrence of \( A \):

   \[
   E(Y \mid A) = \frac{E(Y \cdot 1_A)}{P(A)}.
   \]

   This step ensures that the expectation is only considering the cases where \( A \) has occurred, and scales the result by \( P(A) \) to represent the "average" conditional on \( A \).

Thus, the given formula holds:

\[
E(Y \mid A) = \frac{E(Y \cdot 1_A)}{P(A)}.
\]

This is the required proof for the conditional expectation formula.

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Would you like more details or have any questions on this proof?

Here are some related questions for further exploration:

1. How is the concept of conditional expectation extended to continuous random variables?
2. What is the difference between conditional expectation and conditional probability?
3. Can you provide examples of using this formula in statistical applications?
4. How does this formula relate to the law of total expectation?
5. What happens when \( P(A) = 0 \) in the formula?

**Tip:** When using conditional expectations in probability, always ensure that the event you condition on has a non-zero probability to avoid undefined expressions.

Explore the proof of the conditional expectation formula E(Y|1_A) = E(Y·1_A) / P(A) using indicator functions. This solution includes key mathematical concepts like probability, conditional expectation, and the use of indicator functions. Suitable for advanced high school and undergraduate students.

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