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After reading this and this thread, I got a little confused about the distinction between the Bernoulli and the binomial distribution in practice.

From what I read, I understood that:

  1. A trial (e.g. flipping a coin) consisting of $k$ repetitions, is considered Bernoulli distributed.
  2. When trials are repeated $n$ times, the $n \times k$ repetitions are binomially distributed.

The first thread says the following: "A Bernoulli random variable has two possible outcomes: 0 or 1. A binomial distribution is the sum of independent and identically distributed Bernoulli random variables."

Usually, a coin is used as an example, which means that the data come from the same entity. Now I'm wondering: If I have one sample of $n$ observations from different persons, which distribution should I use? Some thoughts about this:

  1. Are these observations considered the same random variable if the observations come from different persons (or different coins, if you wish)? If so, I would need a Bernoulli trial.
  2. If not, I might be tempted to say that the variables need not be identically distributed (they might have different mean probabilities for example).

Things become more confusing when take the relation with the multinomial distribution and the multivariate Bernoulli distribution into account (e.g. here). These two are closely related. The multivariate Bernoulli distribution models the observations from one person (or coin). If we want to model the observations of $n$ persons (coins), we would need a multinomial distribution on the sums of outcome combinations. Since a multivariate Bernoulli distribution models multiple outcomes from one person only, I would then assume that each person (or coin) is related to a different random variable.

I'm pretty sure that I'm mixing things up. Can someone help me to clarify things?

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