I am trying to find the expected fraction of roots located in the unit disc for a random polynomial with Gaussian coefficients. Given a random polynomial
$$P(z) = a_0 + a_1 z + a_2 z^2 + \dots + a_n z^n $$
where each coefficient ( a_i ) is a real Gaussian random variable, I am interested in understanding how many of the roots of this polynomial, on average, will lie inside the unit disc in the complex plane, i.e., have a magnitude less than 1.
Formally, let $ C_n $ denote the set of roots of the polynomial and $ |C_n| \$denote the number of such roots. I want to find the expected value of the fraction of roots in the unit disc:
$$ \mathbb{E} \left[ \frac{| \{ z \in C_n : |z| < 1 \} |}{|C_n|} \right] $$
Any insights, references, or solutions regarding this problem will be greatly appreciated.
Thank you in advance for your help!