Traditional approaches to the study of random polynomials and random analytic functions have focussed on answering questions regarding the behavior and/or location of zeros of these functions, where the randomness in these functions arises from the choice of coefficients. In this thesis, we shall flip this model - we consider random polynomials and random analytic functions where the source of randomness is in the choice of zeros. While first chapter is devoted to an introduction into the field, in the next two chapters, we consider random polynomials whose zeros are chosen IID using some distribution. The second chapter answers questions regarding the asymptotic distribution of the critical points of a random polynomial whose zeros are IID on a circle on the complex plane. The fourth chapter describes the asymptotic behavior of the coefficients of a random polynomial whose zeros are IID Rademacher random variables. In the third chapter, we consider a random entire function that vanishes at a Poisson point process of intensity 1 on R. We give results on the asymptotic behavior of the coefficients as well as the resulting zero set on repeatedly differentiating this function.