Let us consider the following dataset

$ D = \{d_1, d_2, \cdots, d_k\} $ where $ d_i \in \mathbb{R}^n$ for all $1 \leqslant i \leqslant n$ and $n, k, i \in \mathbb{N}$.

A probability distribution $P$ is a function from powerset of sample space $S$ of a random experiment $E$ to $[0,1]$.

In this context, what does it mean if a dataset is generated by a probability distribution?

Is generation here a philosophical word like for everything that happens in this universe happens with some probability?

Or is there any theorem that states a probability distribution generate generates a dataset or for any possible data set there is some underlying probability distribution?

  • $\begingroup$ Maybe they are referring to the CDF of the distribution and they simply wanted to say that they draw samples from that distribution ? $\endgroup$ – deemel Apr 26 at 6:25

This is the basic assumption for statistics to operate, namely that a sample $(d_1,\ldots,d_n)$ is the realisation of an iid $n$-sample from a probability distribution $P$, ie a random vector of dimension $n$ distributed from $P^{\otimes n}$. This is not necessarily what happens in reality but this modelling is fundamental to use and justify statistical tools. Without a modicum of repeatability and stabilisation like the law of large numbers, statistics cannot operate.


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