# What does it mean by the generation of the dataset by probability distribution?

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?

• Maybe they are referring to the CDF of the distribution and they simply wanted to say that they draw samples from that distribution ? – 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.