# What is the distribution of multiplication of two Gaussian polynomials?

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.

• – Tlön Uqbar Orbis Tertius Feb 2 '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 2 '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 (mathworld.wolfram.com/KacFormula.html).) – Jean Marie Feb 2 '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 2 '17 at 14:39
• "darga"="degree" in which language ? – Jean Marie Feb 2 '17 at 14:41