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Let's analyze each function to determine if it can represent a valid probability function. A valid probability function must satisfy two conditions:

1. **Non-negativity:** \( f(x) \geq 0 \) for all \( x \).
2. **Total probability equals 1:** The sum of \( f(x) \) over all possible values of \( x \) must be equal to 1.

### Part (a)
\( f(x) = \frac{1}{6} \) for \( x = 1, 2, 3, 4 \), and \( f(x) = 0 \) elsewhere.

- **Non-negativity:** \( f(x) = \frac{1}{6} \geq 0 \) for all given \( x \), so it satisfies the non-negativity condition.
- **Sum of probabilities:**  
  \[
  \sum_{x=1}^{4} f(x) = \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} = \frac{4}{6} = \frac{2}{3}
  \]
  Since the total sum is not equal to 1, this is **not** a valid probability function.

### Part (b)
\( f(x) = \left( \frac{1}{4} \right)^x \) for \( x = 1, 2, 3, \ldots \), and \( f(x) = 0 \) elsewhere.

- **Non-negativity:** \( f(x) = \left( \frac{1}{4} \right)^x \geq 0 \) for all \( x \geq 1 \), so it satisfies the non-negativity condition.
- **Sum of probabilities:**  
  The sum of this geometric series is:
  \[
  \sum_{x=1}^{\infty} f(x) = \sum_{x=1}^{\infty} \left( \frac{1}{4} \right)^x = \frac{\frac{1}{4}}{1 - \frac{1}{4}} = \frac{\frac{1}{4}}{\frac{3}{4}} = \frac{1}{3}
  \]
  The sum is \( \frac{1}{3} \), not 1, so this is **not** a valid probability function.

### Part (c)
\( f(x) = (1 - p)p^x \) for \( x = 0, 1, 2, \ldots \), where \( 0 \leq p \leq 1 \), and \( f(x) = 0 \) elsewhere.

- **Non-negativity:** Since \( 0 \leq p \leq 1 \), both \( 1 - p \) and \( p^x \) are non-negative, so \( f(x) \geq 0 \) for all \( x \geq 0 \).
- **Sum of probabilities:**  
  The sum of this geometric series is:
  \[
  \sum_{x=0}^{\infty} f(x) = \sum_{x=0}^{\infty} (1 - p)p^x = (1 - p) \sum_{x=0}^{\infty} p^x = (1 - p) \frac{1}{1 - p} = 1
  \]
  The sum equals 1, so this **is** a valid probability function.

### Conclusion:
Only **(c)** is a valid probability function.

Would you like further clarification or details on any of these steps? 

Here are some related questions to explore:

1. How do geometric series contribute to probability functions?
2. What are the common characteristics of discrete probability distributions?
3. Can you explain why function (b) does not sum to 1?
4. How do you verify a probability mass function (PMF) for different distributions?
5. How can non-standard functions (like in part a) be modified to become valid probability functions?

**Tip:** When working with series sums in probability functions, always verify both the non-negativity and the total sum of probabilities to ensure it's valid.

Which of the following functions are possible probability functions? Justify your answer.

Explore the analysis of different functions to determine their validity as probability functions. This problem covers key concepts in probability, including non-negativity, total probability condition, and geometric series. Suitable for university-level students.

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