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how many probabilistic distributions for statistical data exists in mathematics ?

is the number finite or infinite ?

how could i prove that for big number of data $ N \to \infty $ all the pdf distribution functions are equal to the normal distributions ?

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Is this along this lines of what you're asking?

https://en.wikipedia.org/wiki/Central_limit_theorem

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    yes, this is one of the questions the other is to know how many pdf there are in mathematics :D if the number if finite – 2017-02-17
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    A probability density function is merely a function that maps values a continuous variable can take onto the probability that a random variable would take on that value – 2017-03-01
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    Lifted from [here](https://en.wikipedia.org/wiki/Probability_density_function): In a more precise sense, the PDF is used to specify the probability of the random variable falling within a particular range of values, as opposed to taking on any one value. This probability is given by the integral of this variable’s PDF over that range—that is, it is given by the area under the density function but above the horizontal axis and between the lowest and greatest values of the range. The probability density function is nonnegative everywhere, and its integral over the entire space is equal to one. – 2017-03-01
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    In that sense, there are infinitely many possible PDFs – 2017-03-01