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If all the probabilities are equal (say to $p$), then we call this sequence of random variables a Bernoulli process. But what is the terminology ( name ) for a sequence with different probabilities $\vec{p} = ( p_1 , p_2 , \cdots )$ ? These individual probabilities can be assumed to be independent of each other.

Since there seem to be some confusion about what being asked, I want to clarify that the object I'm referring to is a stochastic process. Some well-known ones are Bernoulli process, Random Walk, Markov process, etc.

From Wikipedia: "In probability theory and related fields, a stochastic or random process is a mathematical object usually defined as a sequence of random variables in a probability space, where the index of the sequence often has the interpretation of time."

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If $p$ is fixed and all the experiments are Bernoulli trials then we can call it a Binomial Distribution.

If $p$ is not fixed, then it is more useful to consider the Bernoulli trials as separate events and just use discrete probability calculations.

Unless the events are somehow related, there will be no overall formula so it is not likely to be given a name.

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It is Poisson Binomial Distribution :

In Probability theory and Statistics, the Poisson Binomial Distribution is the Discrete Probability Distribution of a sum of independent Bernoulli trials that are not necessarily identically distributed.
The concept is named after Siméon Denis Poisson.

In other words, it is the Probability Distribution of the number of Successes in a Collection of n independent yes/no experiments with Success Probabilities $p_{1} , p_{2} , \dots , p_{n}$
The "Ordinary Binomial Distribution" is a special case of the Poisson Binomial Distribution , when all Success Probabilities are the same, that is $p_{1} = p_{2} = \cdots = p_{n}$

Check out :

https://en.wikipedia.org/wiki/Poisson_binomial_distribution

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    $\begingroup$ Thanks - your answer is better than mine. Up-voting it now. $\endgroup$
    – Red Five
    Apr 11 at 5:13

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