Cumulative distributed function example

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Discussion Overview

The discussion revolves around the properties and definitions of cumulative distribution functions (CDF) in relation to a given probability density function (PDF). Participants explore the implications of the definitions and the behavior of the CDF at specific points, particularly for values beyond the support of the PDF.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant questions why the cumulative distribution function F(x) equals 1 for x ≥ 2, given that the probability density function f(x) is 0 for x outside its defined range.
  • Another participant provides a simpler example to illustrate that F(x) represents the probability that x is less than or equal to a certain value, suggesting that F(3) should be 1 because it includes the probabilities from the defined range of f(x).
  • A participant asserts that the probability of x being exactly 3 is 0, leading them to believe that F(3) should not be defined.
  • Another participant challenges this view, clarifying that F(3) is not the probability of x being 3, but rather the cumulative probability up to that point, which includes all values defined in f(x).
  • It is noted that if f(x) is a probability density function, then F(x) can be computed as the integral of f(x) from negative infinity to x.

Areas of Agreement / Disagreement

Participants express differing views on the interpretation of the cumulative distribution function and its behavior at points outside the support of the probability density function. There is no consensus on whether F(3) should be considered defined or not, and the discussion remains unresolved.

Contextual Notes

Participants reference the definitions of cumulative distribution functions and probability density functions, but there are differing interpretations of how these definitions apply at specific values outside the defined range of the PDF.

xeon123
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I was looking to a video about cumulative distribution function () and he show the following function:[itex]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ | 1/4, 0 \leq x \leq1 \\<br /> f(x) =<(x^3)/5, 1 \leq x \leq 2 \\<br /> \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ |0, otherwise.[/itex]

At minute 8:45, he presents the cumulative distribution as:[itex] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ | 0, x \leq 0 \\<br /> F(x) = < \frac{1}{4}x, 0 \leq x \leq 1 \\<br /> \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ | \frac{1}{20}(x^4+4), 1 \leq x \leq 2 \\<br /> \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ | 1, \ x \geq 2[/itex]

I don't understand why F(x) is 1 for [itex]x \geq 2[/itex], if f(x) is 0, otherwise. Why?BTW, I hope that that my functions are legibles, because I don't know how to put big curly brackets.
 
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xeon123 said:
I don't understand why F(x) is 1 for [itex]x \geq 2[/itex], if f(x) is 0, otherwise. Why?

Look at a simpler example. Suppose f(x) = 1/2 when x = 1 or x = 2 and f(x) = 0 otherwise. The value of the cumulative distribution F(x) would be 1 at x = 3 because F(3) gives the probability that x is equal or less than 3. The condition that x is equal or less than 3 includes the cases x = 1 and x = 2.
 
I understand what you said, but the probability of happening 3 is 0, because it's not defined in f(x). For me, F(3) should never be defined.
 
xeon123 said:
F(3) should never be defined.

Do you mean "should" in some moral or religious sense? Mathematics would only care about you opinion if you could show some logical contradiction in the standard definition of cumulative distribution function.

because it's not defined in f(x).

It is defined in the domain of f(x). f(3) = 0. That's part of the "f(x) = 0 otherwise" clause.
 
Last edited:
xeon123 said:
I understand what you said, but the probability of happening 3 is 0, because it's not defined in f(x). For me, F(3) should never be defined.
You seem to be thinking that "F(3)" is the probability that x is equal to 3. That is not the case. F(X) is the probability that x is less than or equal to 3. Since, by the definition of f(x), x must be less than or equal to 2, x therefore must be less than or equal to 3. F(x)= 1 for any number larger than or equal to 2.

If f(x) is the "probability density function" then [itex]F(X)= \int_{-\infty}^X f(x)dx[/itex] is the probability that x is less than or equal to X.
 

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