Consider a random variable $X:\Omega \rightarrow E$ that combines multiple distributions:

$$X(\omega)\sim\begin{cases} N(0,\sigma^2),\hspace{0.2cm} \text{if $\omega =0$}\\ \text{beta}(0,1), \hspace{0.2cm} \text{if $\omega =1$} \end{cases}$$

Is there a concept for a random variable that combines multiple probabilitiy distributions?

A good answer would be of the sort:

Yes these are called [something]-distributions, and are commonly used in sub-field x. A good reference for these is work y.


No, these are not useful because of [reasons], and have no formal treatment as far as I know


The term you are looking for is a mixture distribution.

Mixture distributions get used in situations when:

  • the observations involve a response to a choice (latent variable models),
  • the data generating process has multiple possible paths (e.g. one of N machines processes a request and the machines are not identical), or
  • when there are "structural" reasons to observe certain outcomes.

An example of the latter is that we could measure how much money people have invested. For many people, that amount would be zero while for people who have invested the amount would be a positive real number. The zeros exist and have non-negligible probability because there is a process for someone to opt into investing (probably by having enough money that they can afford to invest).

The Wikipedia page has a little on mixture distributions, but if you really want to read up on mixture modeling, I would read McLachlan and Peel's Finite Mixture Models or Mengersen, Robert, and Titterington's Mixtures: Estimation and Applications.

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