Math Problem Statement
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Solution
The image you uploaded presents a theorem about an absolutely continuous random variable and its cumulative distribution function (CDF) . Here's a detailed explanation of each part of the theorem:
Theorem Explanation:
Let be an absolutely continuous random variable with cumulative distribution function (CDF) . For any pair of real numbers and , where , the following properties hold:
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:
Since is absolutely continuous, the probability that takes any specific value (like ) is zero. This is a key property of continuous random variables, where probabilities are associated with intervals rather than individual points. -
:
The probability that lies in any of the intervals between and (with different combinations of inclusive/exclusive boundaries) is given by the difference between the CDF values at and . Since is continuous, the probabilities are the same regardless of whether the endpoints are included or excluded. -
( \mathbb{P}(a < X < \infty) = 1 - F_X(a):
The probability that is greater than (i.e., ) is equal to 1 minus the CDF value at . This is because gives the probability that is less than or equal to , so the complement (everything greater than ) is . -
( \mathbb{P}(-\infty < X \leq b) = F_X(b):
The probability that is less than or equal to (i.e., ) is simply the value of the CDF at , . This is a standard property of the cumulative distribution function.
Summary:
- The first statement emphasizes that individual points have zero probability for continuous random variables.
- The second statement shows that the probability of lying within any interval between and depends only on the values of the CDF at those points, regardless of whether the endpoints are included.
- The third and fourth statements give the probabilities for intervals extending to infinity or from negative infinity, using the CDF values.
Do you have any questions or need further details on any part of this theorem?
Here are some follow-up questions:
- What is the definition of an absolutely continuous random variable?
- How is the cumulative distribution function related to the probability density function ?
- Can you explain how the CDF is used to calculate probabilities?
- What is the difference between an absolutely continuous and a discrete random variable?
- Why do continuous random variables assign zero probability to individual points?
Tip: The CDF is a powerful tool to compute probabilities, especially when working with continuous random variables.
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Math Problem Analysis
Mathematical Concepts
Probability Theory
Random Variables
Cumulative Distribution Function (CDF)
Formulas
P(X = a) = 0
P(a < X ≤ b) = P(a ≤ X ≤ b) = P(a ≤ X < b) = P(a < X < b) = FX(b) - FX(a)
P(a < X < ∞) = 1 - FX(a)
P(-∞ < X ≤ b) = FX(b)
Theorems
Theorem for Absolutely Continuous Random Variables and Cumulative Distribution Functions
Suitable Grade Level
University Level (Undergraduate Probability and Statistics)
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