Constraints on Distribution Functions

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SUMMARY

The discussion clarifies that a distribution function \( F_X \) must be right continuous, defined as \( F(x) = P(X \leq x) \), to ensure consistency in probability calculations. While left continuity, represented by \( F(x) = P(X < x) \), is also valid, the right continuity is the standard convention. The necessity for non-decreasing behavior and the limit approaching 1 as \( x \) approaches infinity are essential characteristics of valid distribution functions.

PREREQUISITES
  • Understanding of probability theory and distribution functions
  • Familiarity with right and left continuity concepts
  • Knowledge of non-decreasing functions
  • Basic grasp of limits in calculus
NEXT STEPS
  • Research the properties of distribution functions in probability theory
  • Explore the implications of right and left continuity in statistical models
  • Study the concept of non-decreasing functions in mathematical analysis
  • Learn about limits and their applications in defining continuity
USEFUL FOR

Statisticians, mathematicians, and students studying probability theory who need a deeper understanding of distribution functions and their properties.

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Why does a distribution function [tex]F_X[/tex] have to be right continuous ?
is'nt just making sure it is non-decreasing enough
 
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In general, the continuity assumption of distribution functions at jumps is determined by convention. It could be right cont. or left cont. as long as it is used consistently. What you have is a non-zero prob. for the jump value. The main point is the precise definition of F(x) as a distribution function. You can have either of the following:
F(x)=P(X<x) - left continuous.
F(x)=P(X≤x) - right continuous.

Non-decreasing is required as well as the limit =1 at +∞.
 
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