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The content of the image appears to be an exercise in probability theory and focuses on verifying certain properties related to conditional expectations. Specifically, the questions deal with:

1. Listing sets in the sigma algebra \( \sigma(X) \).
2. Determining \( E[Y | X] \), and verifying the partial-averaging property for specific values of \( a \), \( b \), \( c \), and \( d \).
3. Determining \( E[Z | X] \) and verifying the partial-averaging property again.
4. The final problem relates to \( E[Y | X] \) and makes use of properties from conditional expectations, specifically from a theorem.

### Here's a breakdown of the tasks:

- **(i)**: Listing the sets in \( \sigma(X) \), where \( X = \{0, \{a,b\}, \{c,d\}\} \).
- **(ii)**: Calculation of \( E[Y | X] \), demonstrating it for some values of \( a, b, c, \) and \( d \). This also involves verifying the partial-averaging property.
- **(iii)**: A similar verification for \( E[Z | X] \), where \( Z \) is another random variable.
- **(iv)**: There's an explanation of conditional expectation properties, referencing a theorem from conditional expectation theory.

### Would you like a more detailed explanation of any part of this solution, or would you like me to walk you through solving these problems step by step?

Here are five questions that relate to the content of this exercise:

1. What is the partial-averaging property of conditional expectation, and why is it important?
2. How does

List the sets in σ(X), determine E[Y|X] and verify the partial-averaging property, determine E[Z|X], and verify the partial-averaging property.

Learn how to verify the partial-averaging property of conditional expectation, calculate E[Y|X] for specific values of random variables, and explore E[Z|X] using theorems in probability theory. Suitable for advanced undergraduate or graduate students.

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