Math Problem Statement

explIN

Let X be an absolutely continuous random variable, with cumulative distribution function FX, then for any pair (a, b) such that a < b, prove several properties of the probability.

Solution

The image you uploaded presents a theorem about an absolutely continuous random variable XX and its cumulative distribution function (CDF) FXF_X. Here's a detailed explanation of each part of the theorem:

Theorem Explanation:

Let XX be an absolutely continuous random variable with cumulative distribution function (CDF) FXF_X. For any pair of real numbers aa and bb, where a<ba < b, the following properties hold:

  1. P(X=a)=0\mathbb{P}(X = a) = 0:
    Since XX is absolutely continuous, the probability that XX takes any specific value (like aa) is zero. This is a key property of continuous random variables, where probabilities are associated with intervals rather than individual points.

  2. P(a<Xb)=P(aXb)=P(aX<b)=P(a<X<b)=FX(b)FX(a)\mathbb{P}(a < X \leq b) = \mathbb{P}(a \leq X \leq b) = \mathbb{P}(a \leq X < b) = \mathbb{P}(a < X < b) = F_X(b) - F_X(a):
    The probability that XX lies in any of the intervals between aa and bb (with different combinations of inclusive/exclusive boundaries) is given by the difference between the CDF values at bb and aa. Since XX is continuous, the probabilities are the same regardless of whether the endpoints are included or excluded.

  3. ( \mathbb{P}(a < X < \infty) = 1 - F_X(a):
    The probability that XX is greater than aa (i.e., X(a,)X \in (a, \infty)) is equal to 1 minus the CDF value at aa. This is because FX(a)F_X(a) gives the probability that XX is less than or equal to aa, so the complement (everything greater than aa) is 1FX(a)1 - F_X(a).

  4. ( \mathbb{P}(-\infty < X \leq b) = F_X(b):
    The probability that XX is less than or equal to bb (i.e., X(,b]X \in (-\infty, b]) is simply the value of the CDF at bb, FX(b)F_X(b). 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 XX lying within any interval between aa and bb 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:

  1. What is the definition of an absolutely continuous random variable?
  2. How is the cumulative distribution function FXF_X related to the probability density function fXf_X?
  3. Can you explain how the CDF FX(b)F_X(b) is used to calculate probabilities?
  4. What is the difference between an absolutely continuous and a discrete random variable?
  5. 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)