Elliott and Halberstam proved that $\sum_{p<n} 2^{ω(n-p)}$ is asymptotic to $φ(n)$. In analogy to the Erdős--Kac Theorem, Elliott conjectured that if one restricts the summation to primes $p$ such that $ω(n-p)\le 2 \log \log n+λ(2\log \log n)^{1/2}$ then the sum will be asymptotic to $φ(n)\int_{-\infty}^λ e^{-t^2/2}dt/\sqrt{2π}$. We show that this conjecture follows from the Bombieri--Vinogradov Theorem. We further prove a related result involving Poisson--Dirichlet distribution, employing deeper lying level of distribution results of the primes.