Given $x$ and $y$ in $\mathbb{R}$, and let $\mathcal{H} = \{ h \mid \mathbb{R} \to \mathbb{N} \}$ be a family of hash functions where $ h(x) = \left\lfloor x + \sum^C_{i=1} U_i \right\rfloor$ for some constant $C$. Morever, $U_i$ is independently and uniformly choosen at random from [0,1].

What is the probability that $h_1(x) = h_2(y)$ occurs, where $h_1$ and $h_2$ are choosen independently at random from $\mathcal{H}$? I know that it looks similar to a *hat kernel* for $C=5$, see Plot (scaled to $[0,1]$). Moreover the distribution of the sum is an Irwin-Hall distribution. Any suggestions how to proceed?

I posted the question on math.stackexchange a week ago, but did not get an answer.