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To show that \( p(x, y \mid z) = p(x \mid z) p(y \mid z) \) when \( x \) and \( y \) are conditionally independent given \( z \), we can proceed step by step as follows:

### Step 1: Definition of Conditional Independence
Conditional independence between \( x \) and \( y \) given \( z \) means that the occurrence of \( x \) does not affect the probability of \( y \) happening, given that \( z \) has occurred. Mathematically, this is expressed as:
\[
p(x \mid y, z) = p(x \mid z)
\]
Similarly, you can write:
\[
p(y \mid x, z) = p(y \mid z)
\]

### Step 2: Joint Probability and Conditional Probability
The joint probability \( p(x, y \mid z) \) can be written as:
\[
p(x, y \mid z) = p(x \mid y, z) \cdot p(y \mid z)
\]
This is a direct application of the chain rule of probability.

### Step 3: Apply Conditional Independence
Using the conditional independence \( p(x \mid y, z) = p(x \mid z) \), substitute this into the equation:
\[
p(x, y \mid z) = p(x \mid z) \cdot p(y \mid z)
\]
This result shows that the joint conditional probability \( p(x, y \mid z) \) factors into the product of the individual conditional probabilities \( p(x \mid z) \) and \( p(y \mid z) \).

### Conclusion
Thus, we have proven that:
\[
p(x, y \mid z) = p(x \mid z) \cdot p(y \mid z)
\]
when \( x \) and \( y \) are conditionally independent given \( z \).

Would you like further details or have any questions?

Here are 5 relative questions to consider:
1. What is the significance of conditional independence in probability theory?
2. How would the result change if \( x \) and \( y \) were not conditionally independent given \( z \)?
3. Can you give an example where conditional independence is used in real-world scenarios?
4. How does the chain rule of probability relate to Bayes' Theorem?
5. What other properties follow from the assumption of conditional independence?

**Tip:** Always ensure to clearly state the assumptions like conditional independence when solving probability problems, as they simplify the derivation process significantly.

Show that p(x, y | z) = p(x | z) p(y | z) when x and y are conditionally independent, given z.

Learn how to prove that p(x, y | z) = p(x | z) * p(y | z) when x and y are conditionally independent given z. This step-by-step solution covers key probability concepts and applies the chain rule of probability. Suitable for undergraduate students.

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