In this paper, we consider the distribution of primes in intervals of length $\asymp\log x$; this leads to insights concerning Poisson random variables. Later, we consider reduced residues $({\rm mod}\,q)$ in short intervals, which similarly motivates us to derive further information concerning moments of binomial random variables. This discussion is extended further, where we consider the distribution of primes in intervals of length $\asymp x^{\theta}$ where $0<\theta<1$. Our analysis gives rise to a family of polynomials.