Constraints on Distribution Functions

In summary, the continuity assumption of a distribution function F_X at jumps is determined by convention and can be either right continuous or left continuous as long as it is used consistently. The main point is the precise definition of F(x) as a distribution function, with the requirements of non-decreasing and a limit of 1 at positive infinity.
  • #1
cappadonza
27
0
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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  • #2
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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What are distribution functions?

Distribution functions are mathematical functions that describe the probability of a random variable taking on a specific value or falling within a specific range of values. They are used to model and analyze data in fields such as statistics, physics, and economics.

What types of constraints can be placed on distribution functions?

Constraints on distribution functions can include restrictions on the range of values that the random variable can take, limitations on the shape of the distribution, and requirements for the distribution to satisfy certain properties or relationships with other variables.

Why are constraints important in distribution functions?

Constraints can help to simplify and make more meaningful the analysis of data using distribution functions. They can also be used to ensure that the distribution accurately represents the underlying process or phenomenon being studied.

How are constraints on distribution functions determined?

Constraints can be determined through theoretical considerations, such as the assumptions made about the data or the model being used, or through empirical methods, such as fitting the distribution to observed data and evaluating the goodness of fit.

What are some common techniques for dealing with constraints on distribution functions?

Some common techniques for dealing with constraints on distribution functions include transformation of the data or the use of specialized distributions that already incorporate the desired constraints. In some cases, constraints can also be incorporated into the modeling process through the use of Bayesian methods.

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