What is the difference between the Poisson process, the Bernoulli process, and normal renewal?

Poisson process is a continuous version of Bernoulli process.

The main difference between the Bernoulli process and Poisson Process

1. In Bernoulli process, inter-arrival times have geometric distribution, on the other hand in Poisson process inter-arrival times have exponential distribution. (Exponential is the continuous analogue of the geometric distribution)
2. The number of arrivals in an interval has binomial distribution in Bernoulli process wheres in Poisson distribution it has Poisson distribution.
3. The arrival times have negative binomial distributions in the Bernoulli process, but in the Poisson process it is Gamma distribution.

Note: Both Binomial and Poisson distribution used to measure number of certain random events (or “successes”) within a time interval. Binomial is based on discrete events and Poisson is based on continuous events.

Given a Binomial distribution with some n,p where n is the number of attempt and p is the probability of success

if n→ and p→0 in such a way that np→λ then that distribution approaches a Poisson distribution with parameter λ.

*** Renewal process is a special kind of arrival process., in which the inter-arrival intervals are positive, independent and identically distributed (IID) random variables. The Poisson process is a special kind of renewal process.

The Poisson process has extra property than Renewal process- Memory-less property.

P(X>t+x | X> t)=P(X>x), x>=0 means it does not depend on past events.

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