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The question is asking about the correct relationship between the probability mass function (pmf) \( p(x) \) and the cumulative distribution function (CDF) \( F(x) \) for a discrete random variable \( X \).

### Key Concepts:
1. **\( p(x) \)**: This is the probability mass function, meaning it gives \( P(X = x) \), the probability that \( X \) takes the specific value \( x \).
2. **\( F(x) \)**: This is the cumulative distribution function, which gives the probability that \( X \) is less than or equal to \( x \), i.e., \( F(x) = P(X \leq x) \).
3. For a discrete random variable, \( F(x) \) is a sum of the probabilities up to \( x \), i.e., \( F(x) = \sum_{t \leq x} p(t) \).

### Evaluating the Options:
- **\( p(x) \geq F(x) \)**: This is **false** because the pmf \( p(x) \) represents a single value's probability, while \( F(x) \) is the cumulative sum of probabilities up to \( x \). Hence, \( F(x) \geq p(x) \) for all \( x \).
  
- **\( p(x) \) and \( F(x) \) can be greater than 1**: This is **false** because both \( p(x) \) and \( F(x) \) represent probabilities, which cannot exceed 1.
  
- **\( F(x) \geq p(x) \)**: This is **true** because the CDF sums all probabilities up to \( x \), so it is always greater than or equal to the probability for a specific \( x \), \( p(x) \).
  
- **\( p(x) = P(X = x) \) and \( F(x) = P(X = x) \) (they mean the same thing)**: This is **false**. The pmf \( p(x) \) gives the probability of \( X \) being exactly \( x \), while the CDF \( F(x) \) gives the probability of \( X \) being less than or equal to \( x \).
  
- **\( p(x) = P(X = x) \) and \( F(x) = P(X > x) \)**: This is **false**. \( F(x) \) gives the probability of \( X \leq x \), not \( P(X > x) \).
  
- **\( F(x) \) and \( p(x) \) can be negative**: This is **false** because probabilities, by definition, cannot be negative.

### Correct Answer:
- The true statement is: **\( F(x) \geq p(x) \)**.

Would you like further details or explanations on any part of this? Here are some additional questions you might find helpful:

1. What is the difference between a probability mass function (pmf) and a probability density function (pdf)?
2. How is the cumulative distribution function (CDF) used in real-world applications?
3. What are the properties of the CDF for a discrete random variable?
4. Can the CDF ever decrease, and if so, under what conditions?
5. How would the relationships between pmf and CDF change for a continuous random variable?

**Tip:** The sum of all probabilities for a discrete random variable, i.e., \( \sum p(x) \), always equals 1, which is a useful property in solving probability problems!

Select the true statement below if X is a discrete random variable. We know that p(x) is the probability mass function (pmf), and F(x) is the cumulative distribution function (CDF).

Explore the relationships between the probability mass function (PMF) and cumulative distribution function (CDF) for a discrete random variable. This problem tests understanding of key concepts in probability theory, including the properties of CDF and PMF.

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