Probability density function

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Discussion Overview

The discussion centers on the properties of convolution in the context of probability density functions (PDFs), specifically whether the convolution of two normalized PDFs results in another normalized PDF. The scope includes theoretical aspects and mathematical reasoning related to normalization in probability theory.

Discussion Character

  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant asserts that the convolution of two normalized probability density functions, g and f, is also normalized, as it represents the density function for the sum of the corresponding random variables.
  • Another participant questions how to demonstrate that normalization is preserved through convolution.
  • A subsequent reply proposes a method to show normalization preservation by expressing the convolution as an integral and applying a change of variables, suggesting that the product of integrals involved equals one.

Areas of Agreement / Disagreement

There is no consensus on the preservation of normalization through convolution, as participants are exploring the proof and reasoning behind the claim.

Contextual Notes

The discussion involves assumptions about the properties of integrals and the definitions of normalized probability density functions, which may not be fully articulated or resolved.

aaaa202
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If g and f are two normalized probability density functions is it then true in general that the convolution of f and g is normalized too?
 
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Yes. The convolution is the density function for the sum of the random variables which have g and f as density functions.
 
how do you show that normalization is preserved..
 
aaaa202 said:
how do you show that normalization is preserved..

Let f(x) = g(x)*h(x) (where * means convolution). ∫f(x)dx = ∫g(x)dx∫h(y-x)dy. Let u = y-x for the h integral (remember these integrals are over the entire real line) and you have the product of 2 integrals, each of which is 1.
 

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