Math Problem Statement

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).

Solution

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

Key Concepts:

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

Evaluating the Options:

  • p(x)F(x)p(x) \geq F(x): This is false because the pmf p(x)p(x) represents a single value's probability, while F(x)F(x) is the cumulative sum of probabilities up to xx. Hence, F(x)p(x)F(x) \geq p(x) for all xx.

  • p(x)p(x) and F(x)F(x) can be greater than 1: This is false because both p(x)p(x) and F(x)F(x) represent probabilities, which cannot exceed 1.

  • F(x)p(x)F(x) \geq p(x): This is true because the CDF sums all probabilities up to xx, so it is always greater than or equal to the probability for a specific xx, p(x)p(x).

  • p(x)=P(X=x)p(x) = P(X = x) and F(x)=P(X=x)F(x) = P(X = x) (they mean the same thing): This is false. The pmf p(x)p(x) gives the probability of XX being exactly xx, while the CDF F(x)F(x) gives the probability of XX being less than or equal to xx.

  • p(x)=P(X=x)p(x) = P(X = x) and F(x)=P(X>x)F(x) = P(X > x): This is false. F(x)F(x) gives the probability of XxX \leq x, not P(X>x)P(X > x).

  • F(x)F(x) and p(x)p(x) can be negative: This is false because probabilities, by definition, cannot be negative.

Correct Answer:

  • The true statement is: F(x)p(x)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., p(x)\sum p(x), 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