I have a set of users that generate calls. If I assign the same probability to each user, they have identical call generation probability which can be defined as $\delta$. These callers are chosen uniformly among the set of users. At the end of the generation process, the representation of the probability density function of the call rates should be a delta function (hence the shape is similar to a bell, isn't it?)
The probability i assigned to each user is: $p_u = \frac{\lambda}{\sum_{i \in N_u} \lambda}$
where $\lambda = \frac{1}{N_u}$ and $N_u$ is the number of users. In this way they are equally partitioned between 0 and 1 and i can take a random number uniformly distributed in order to select a random user.
My question is how can i demonstrate that this is really a Delta function? The information i wrote are enough to defined the Delta function (i don't know if it is possible to formalize the p.d.f.)?
For example in figure we have 10000 that has the same generation probability: if I generate ca. 605000 calls the average is ca. 60.5 calls per user