There are certain probability spaces which I regard as very basic and very easy to match with physical intuition. These are for example:
- The Bernoulli $\mathrm{Ber}_p$ distribution with paramter $p$ on $\Omega= \{0,1\}$.
- The Laplace distribution on $\Omega = \{1, \dots, n\}$.
- The Lebesgue measure on $\Omega = [0,1]$.
I know how a few of the standard discrete probability measures can be constructed from the above (in particular the Bernoulli distribution).
For example, if I consider the product probability space $(\{0,1\}^n, P(\{0,1\}^n), \mathrm{Ber}_p^{\otimes n} )$, consider the sum map $\Sigma \colon \{0,1\}^n \to \{0, \dots, n\}$, then the image measure $\Sigma_*( \mathrm{Ber}_p^{\otimes n})$ is the binomial distribution $\mathrm{Bin}_{n,p}$.
Or, if I consider the infinite product $(\{0,1\}, P(\{0,1\}), \mathrm{Ber}_p)^\mathbb{N}$ and consider the map $X \colon \{0,1\}^\mathbb{N} \to \mathbb{N} \cup \{\infty\}$, $(a_1, a_2, \dots) \mapsto \inf\{n \in \mathbb{N} \mid a_n = 1\}$, then the image measure $X_*(\mathrm{Ber}_p^{\otimes \mathbb{N}})$ is the geometric distribution with parameter $p$.
The above constructions are directly related with real world interpretations of the respective distributions.
I wonder if I can in a similar way construct continuous distributions like the Poisson distribution, the exponential distribution or the Gauß distribution from more basic ones (perhaps using the Lebesgue measure as a basic building block).