To prove the linearity of the conditional expectation, we want to show that for random variables $Y$ and $Z$, and constants $a$ and $b$:

[ \bE[aY + bZ \mid X] = a \bE[Y \mid X] + b \bE[Z \mid X]. ]

Step-by-step Proof

1. Definition of Conditional Expectation: The conditional expectation $\bE[W \mid X]$ for a random variable $W$ is a random variable such that for any measurable set $A$: $\int_A \bE[W \mid X] \, dP = \int_A W \, dP.$

2. Set Up the Equation to Prove: Let $W = aY + bZ$. We need to show that: [ \bE[W \mid X] = a \bE[Y \mid X] + b \bE[Z \mid X]. ]

3. Applying Linearity of Expectations Given $X$: Using the linearity of expectation, which states that for any two random variables $Y$ and $Z$ and constants $a$ and $b$: [ \bE[aY + bZ \mid X] = a \bE[Y \mid X] + b \bE[Z \mid X]. ] This property holds because the expectation operator itself is linear, and the conditioning on $X$ does not affect this linearity.

4. Verification by Properties of Integrals: To ensure that this equality is indeed satisfied, we can test it by checking if the integral conditions hold for both sides. For any measurable function $g(X)$: $\int g(X) \left( \bE[aY + bZ \mid X] \right) dP = \int g(X) (aY + bZ) \, dP.$ Using the linearity of integrals: [ \int g(X) \left( a \bE[Y \mid X] + b \bE[Z \mid X] \right) dP = a \int g(X) Y , dP + b \int g(X) Z , dP. ] Both expressions above are indeed equivalent, which confirms that: [ \bE[aY + bZ \mid X] = a \bE[Y \mid X] + b \bE[Z \mid X]. ]

Conclusion

This completes the proof that the conditional expectation operator is linear: [ \bE[aY + bZ \mid X] = a \bE[Y \mid X] + b \bE[Z \mid X]. ]

Would you like more details on any part of this proof or have any questions about conditional expectations?

Here are some related questions to expand your understanding:

1. What are the key properties of conditional expectations that make this proof possible?
2. How does the concept of measurability play a role in conditional expectations?
3. Can you prove the law of iterated expectations using these properties?
4. How do conditional expectations relate to probability distributions?
5. What is the significance of conditional variance, and how is it related to conditional expectation?

Tip: Always remember that linearity of expectation holds whether or not the random variables involved are independent.