Suppose given two $n$-degree polynomials $p_1$ and $p_2$. Each coefficient of $p_1$ and $p_2$ is independently sampled from Gaussian distribution with mean 0 and standard deviation $\sigma$, i.e. we sample $n$ times from Gaussian distribution and these sampled values are coefficients of $p_1$. Same goes to $p_2$.

I want to know the distribution of $p$=$p_1\cdot$$p_2$. Does $p$ comply with Gaussian distribution? If so, what is the standard deviation? Are there proofs for this question?

Thank you very much.

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  • Related : http://math.stackexchange.com/questions/101062/is-the-product-of-two-gaussian-random-variables-also-a-gaussian – Tlön Uqbar Orbis Tertius Feb 02 '17 at 14:29
  • By giving this reference, @Tlön Uqbar Orbis Tertius implicitely means that already with constant polynomials, it isn't true. – Jean Marie Feb 02 '17 at 14:33
  • Otherwise, the seminal works done on the subject of random polynomials in the 1950's are by Mark KAC (see for example (http://mathworld.wolfram.com/KacFormula.html).) – Jean Marie Feb 02 '17 at 14:36
  • By the way, reasoning on the distribution of zeros, would maybe another angle of attack of the issue... – Jean Marie Feb 02 '17 at 14:39
  • "darga"="degree" in which language ? – Jean Marie Feb 02 '17 at 14:41
  • @JeanMarie Thank you. I found this [paper](https://pdfs.semanticscholar.org/2e96/6bde2e2fbef6fb9324d425909642bc5bbd05.pdf). It says 'The product of two symmetric and unimodal polynomials of dargas m and m' respectively is a symmetric and unimodal polynomial of darga nz + m'. ' on page 2 observation 2. Do you think this is related to my question? – 9f241e21 Feb 02 '17 at 14:41
  • @JeanMarie ok. They are not the same thing. – 9f241e21 Feb 02 '17 at 14:42
  • @JeanMarie But for my question, I think darga should equals to degree – 9f241e21 Feb 02 '17 at 14:43
  • Indeed, the paper is combinatorics-oriented. No probability. – Jean Marie Feb 02 '17 at 14:45

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