Math Problem Statement
Solution
The question is asking about the correct relationship between the probability mass function (pmf) and the cumulative distribution function (CDF) for a discrete random variable .
Key Concepts:
- : This is the probability mass function, meaning it gives , the probability that takes the specific value .
- : This is the cumulative distribution function, which gives the probability that is less than or equal to , i.e., .
- For a discrete random variable, is a sum of the probabilities up to , i.e., .
Evaluating the Options:
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: This is false because the pmf represents a single value's probability, while is the cumulative sum of probabilities up to . Hence, for all .
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and can be greater than 1: This is false because both and represent probabilities, which cannot exceed 1.
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: This is true because the CDF sums all probabilities up to , so it is always greater than or equal to the probability for a specific , .
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and (they mean the same thing): This is false. The pmf gives the probability of being exactly , while the CDF gives the probability of being less than or equal to .
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and : This is false. gives the probability of , not .
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and can be negative: This is false because probabilities, by definition, cannot be negative.
Correct Answer:
- The true statement is: .
Would you like further details or explanations on any part of this? Here are some additional questions you might find helpful:
- What is the difference between a probability mass function (pmf) and a probability density function (pdf)?
- How is the cumulative distribution function (CDF) used in real-world applications?
- What are the properties of the CDF for a discrete random variable?
- Can the CDF ever decrease, and if so, under what conditions?
- 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., , always equals 1, which is a useful property in solving probability problems!
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Math Problem Analysis
Mathematical Concepts
Probability Theory
Discrete Random Variables
Probability Mass Function
Cumulative Distribution Function
Formulas
F(x) = P(X ≤ x)
p(x) = P(X = x)
F(x) = Σp(t) for t ≤ x
Theorems
Cumulative Distribution Function Properties
Probability Mass Function Properties
Suitable Grade Level
University level
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