Binomial vs Poisson
Despite the fact, numerous distributions fall in the category of ‘Continuous Probability Distributions’ Binomial and Poisson set examples for the ‘Discrete Probability Distribution’ and among widely used as well. Beside this common fact, significant points can be brought forward to contrast these two distributions and one should identify at which occasion one of this has been rightly chosen.
Binomial Distribution
‘Binomial Distribution’ is the preliminary distribution used to encounter, probability and statistical problems. In which a sampled size of ‘n’ is drawn with replacement out of ‘N’ size of trials out of which yields a success of ‘p’. Mostly this has been carried out for, experiments which provides two major outcomes, just like ‘Yes’, ‘No’ results. On the contrary to this, if the experiment is done without replacement, then model will be met with ‘Hypergeometric Distribution’ that to be independent from its every outcome. Though ‘Binomial’ comes into play at this occasion as well, if the population (‘N’) is far greater compared to the ‘n’ and eventually said to be the best model for approximation.
However, at most of the occasions most of us get confused with the term ‘Bernoulli Trials’. Nevertheless, both the ‘Binomial’ and ‘Bernoulli’ are similar in meanings. Whenever ‘n=1’ ‘Bernoulli Trial’ is especially named, ‘Bernoulli Distribution’
The following definition is a simple form of bringing the exact picture between, ‘Binomial’ and ‘Bernoulli’:
‘Binomial Distribution’ is the sum of independent and evenly distributed ‘Bernoulli Trials’. Below mentioned are some important equations comes under category of ‘Binomial’
Probability Mass Function (pmf): (nk) pk(1-p)n-k ; (nk) = [n !] / [k !] [(n-k) !]
Mean: np
Median: np
Variance: np(1-p)
At this particular example,
‘n’- The whole population of the model
‘k’- Size of the which is drawn and replaced from ‘n’
‘p’- Probability of success for every set of experiment which consists only two outcomes
Poisson Distribution
On the other hand this ‘Poisson distribution’ has been chosen at the event of most specific ‘Binomial distribution’ sums. In other words, one could easily say that ‘Poisson’ is a subset of ‘Binomial’ and more of a less a limiting case of ‘Binomial’.
When an event occurs within a fixed time interval and with a known average rate then it is common that case can be modeled using this ‘Poisson distribution’. Besides that, the event must be ‘independent’ as well. Whereas it is not the case in ‘Binomial’.
‘Poisson’ is used when problems arise with ‘rate’. This is not always true, but more often than not it is true.
Probability Mass Function (pmf): (λk /k!) e-λ
Mean: λ
Variance: λ
What is the difference between Binomial and Poisson?
As a whole both are examples of ‘Discrete Probability Distributions’. Adding to that, ‘Binomial’ is the common distribution used more often, however ‘Poisson’ is derived as a limiting case of a ‘Binomial’.
According to all these study, we can arrive at a conclusion saying that regardless of the ‘Dependency’ we can apply ‘Binomial’ for encounter the problems as it is a good approximation even for independent occurrences. In contrast, the ‘Poisson’ is used at questions/problems with replacement.
At the end of the day, if a problem is solved with both the ways, which is for ‘dependent’ question, one must find the same answer at each instance.