Explaining the concept of having data distributed in some probability distribution Cordially, can someone explain, in a plain simple way, what do we mean by saying "input data are distributed according to some probability distribution"
Possibly this question sounds trivial to the community here but in fact I've never been able to understand it. I truly appreciate any answer.
 A: Suppose your data is $x_1,x_2,\ldots$, taking value in $\{0,1,\ldots\}$ for instance. And let Y be probability distribution, for instance the Poisson distribution with parameter $\lambda$. $$Y \leadsto Poi(\lambda)$$
For any $k$, the probability that $Y$ is $k$ is defined by
$$\mathbb{P}(Y=k) = e^{-\lambda}\frac{\lambda^k}{\lambda !}$$
We say that your data follow this Poisson distribution if, if you take (uniformly) at random an element $x$ from $x_1,x_2,\ldots$, the probability that $x$ is equal to $k$ (for any $k$) corresponds to the Poisson probability : 
$$\mathbb{P}(x=k) = e^{-\lambda}\frac{\lambda^k}{\lambda !}$$
This is of course an idealised view. In reality, a data set will only approach this value. That's why there exist some statistical tests allowing to decide if some data follow a distribution.
Simpler example: suppose your data is composed of $0$ and $1$. We could say that your data follow a Bernouilli distribution with parameter $p$ if $100p$% of your data is equal to $0$, while $100-p$% of your data is equal to $1$. 
