5

The process is defined similarly to the Bernoulli process composed of $n$ Bernoulli trials. The difference is that the trials are dependent, that is:

$$ P(X_i = 1 | X_1, ..., X_{i-1}) = \frac{m -\sum_{j = 1}^{i - 1} X_j}{m} p , $$

where m is a natural number.

Ben
  • 91,027
  • 3
  • 150
  • 376
abukaj
  • 353
  • 1
  • 11
  • @AdamO I agree. But in this case I guess the process as a whole has no longer such Bernoulli process properties as binomial distribution of number of successes. – abukaj Feb 14 '18 at 17:45
  • @AdamO But the whole process is not a Bernoulli process. According to Wikipedia _"Bernoulli process is a sequence of **independent identically distributed** Bernoulli trials"_ (emphasis mine) – abukaj Feb 14 '18 at 18:09
  • @AdamO May you provide me with a quotable source (that it is a BP)? That is different from what I have been taught so far. I guess I need to provide some reference in my thesis then in case my referees were taught the same. – abukaj Feb 14 '18 at 18:20
  • I think I was confusing discrete time stochastic processes with Bernoulli process. Thanks for encouraging me to dive into notes. I think "Discrete Time Process" is a general framework that can describe the type of sequence you are describing. The probability model, as I said, could be a special case of some autoregressive structure of which I'm unaware. Other than that, I doubt there is a specific title for the probability model you describe. – AdamO Feb 14 '18 at 18:26
  • 2
    This looks like an urn problem. Assume you have two urns, one (#1) with $m$ blue balls and no red balls and the other (#2) with a single red ball and no blue balls. You draw from urn 1 with probability $p$ and replace whichever ball is drawn with a red ball. If you draw from urn 2 you replace the red ball with another red ball (thus maintaining a single red ball in urn 2 at all times)... – jbowman Feb 14 '18 at 22:16
  • You can see how this would generalize nicely to $(m_i, n_i)$ (blue, red) balls in each urn. Here we have $(m, 0)$ (blue, red) balls in urn 1 and $(0, n)$ (blue, red) balls in urn 2, where I've arbitrarily set $n=1$ in the example above. The process stops evolving when all the blue balls are gone, i.e., in state $((0, m), (0, n))$. I don't know of a name for an urn model with this structure, though, but I'm hardly an expert in this area. – jbowman Feb 14 '18 at 22:16

0 Answers0