Probability density function of digital filter

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Discussion Overview

The discussion revolves around determining the probability density function (pdf) of a variable defined as a function of independent exponential random variables. The scope includes theoretical aspects of probability distributions and convolution of pdfs.

Discussion Character

  • Technical explanation, Mathematical reasoning, Debate/contested

Main Points Raised

  • Some participants note that if x has an exponential density function, the pdf of y(n) defined as the sum of two independent variables can be obtained through convolution of their respective pdfs.
  • One participant claims that y(n) will be distributed as gamma(2,1) if x(n) has the pdf exp[-x(n)].
  • A later reply corrects the earlier definition of y(n) to y(n) = [x(n-1) + x(n)]^2 and questions whether the pdf of y(n) is the convolution of exp(-x(n)^2) and exp(-x(n-1)^2).

Areas of Agreement / Disagreement

Participants express differing views on the correct formulation of y(n) and its implications for the pdf, indicating that the discussion remains unresolved.

Contextual Notes

There are limitations in the assumptions regarding the independence of the variables and the specific forms of the pdfs being discussed, which may affect the conclusions drawn.

purplebird
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given that x has an exponential density function ie p(x) = exp (-x) and x(n) & x(m) are statistically independent.

Now y(n) = x(n-1)+x(n)

what is the pdf (probability density function) of y(n)
 
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purplebird said:
given that x has an exponential density function ie p(x) = exp (-x) and x(n) & x(m) are statistically independent.

Now y(n) = x(n-1)+x(n)

what is the pdf (probability density function) of y(n)

The pdf of a sum of two independent variables can be obtained by the convolution of their respective pdf's.
 
Y(n) will be distributed as gamma(2,1) if X(n) has the pdf exp[-x(n)].
 
I made a mistake while typing up the question :

y(n) = [x(n-1) + x(n)]^2

so

y(n) = x(n)^2 + x(n-1)^2

So is the pdf of y(n) convolution of exp(-x(n)^2) and exp(-x(n-1)^2)

Thanks
 

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