Consider a polynomial $p(z)= \sum_0^n a_i z^i.$ In the literature there are numerous bounds about the roots of $p(z)$.Now once we prescribe certain dsitribution to the coefficients ,the bound itself becomes a random variable.Generally ,it might not be always possible to find the distribution of the bound .Suppose in certain cases we are able to find the distribution of the bound,say it is $F(y).$Then can we assert the following :
For a positive number $c$,all the roots are less or equal to $c$ with a probability that is at least equal to $F(c)$? I would be highly obliged for any help/clarification in this regard.