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It is possible to define a simple discrete random variable in terms of Bernoulli distributions? If the answer to this question is so long or broad I would like to know some bibliography about the topic.

To put my question in a clear context: I have a discrete random variable $X$, that can take values in a finite set $\{z_1,\ldots,z_2\}$ for $z_1,z_2\in\Bbb Z$ (with some PMF $f_X$ defined by a function that maps $z_1\mapsto p_1$ with $p_j\in[0,1]$ and $\sum_j p_j=1$).

Then I want to know if $f_X$ can be written as a composition of Bernoulli random variables, by example as a polynomial of Bernoulli random variables or something similar. Thank you.

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I think what you're looking for is the Categorical distribution:

A formulation that appears complex but facilitates mathematical manipulations is as follows, using the Iverson bracket:

$$f(x\mid p)=\prod_{i=1}^{k}p_{i}^{[x=i]},$$ where [x=i] evaluates to 1 if x=i, 0 otherwise, and $p=(p_1,\ldots,p_n)$. There are various advantages of this formulation, e.g.:

It is easier to write out the likelihood function of a set of independent identically distributed categorical variables.

It connects the categorical distribution with the related multinomial distribution.

It shows why the Dirichlet distribution is the conjugate prior of the categorical distribution, and allows the posterior distribution of the parameters to be calculated.

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