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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 enter image description here

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    Do you know what a delta function *is*?2011-08-02
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    good question, this is a new field for me, i obviously read different article but this delta made me very confused. E.g. "delta function is not really a function but it is used also as distribution" etc . And i thought that it is better to understand the delta through the mentioned example.2011-08-02
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    Visually, a delta function should look like an infinitesimally thin, infinitely tall [spike](http://upload.wikimedia.org/wikipedia/commons/4/48/Dirac_distribution_PDF.svg), which represents the fact a random variable can only take on one specific value with 100% probability. What you have looks like a [Binomial distribution](http://upload.wikimedia.org/wikipedia/commons/b/b7/Binomial_Distribution.svg), which is approximately a normal distribution, hence the bell-shape.2011-08-02
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    thank you, yes it looks like a normal distribution. But the source code of the application that generates this calls does not take into account any standard deviation or something like that. I read that sometimes the delta is considered as a sequence of Gaussian (but i don't understand very well) and maybe it can look not so infinitesimally thin.2011-08-02
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    I would say, the question should be: **Is this a normal distribution?**2011-08-02
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    can someone explain me something about the fact that the Delta function is considered as a limit of Gaussian?2011-08-02
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    @Maurizio: You might try [Khan Academy](http://www.khanacademy.org/video/dirac-delta-function?playlist=Differential%20Equations) (I've heard good things about it, though I've never looked at it myself). Do understand that your situation does not involve the delta function, though.2011-08-02

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