If both the Poisson and Binomial distribution are discrete, then why do we need two different distributions?
3$\begingroup$ There are many more than 2 discrete distributions. Poisson counts the number of occurrences in an interval given a certain average occurence rate per interval. It can have values 0, 1, 2, 3, ... Binomial counts the number of occurrences out of a fixed number N of possibilities, where any one occurrence happens with probability p. It can take values 0, 1, 2...N. Quite different situations. $\endgroup$– PaulDec 3, 2014 at 17:28
3$\begingroup$ If both apples and oranges are fruits, why do we need both? $\endgroup$– user147263Dec 3, 2014 at 20:14
The Binomial and Poisson distributions are similar, but they are different. Also, the fact that they are both discrete does not mean that they are the same. The Geometric distribution and one form of the Uniform distribution are also discrete, but they are very different from both the Binomial and Poisson distributions.
The difference between the two is that while both measure the number of certain random events (or "successes") within a certain frame, the Binomial is based on discrete events, while the Poisson is based on continuous events. That is, with a binomial distribution you have a certain number, $n$, of "attempts," each of which has probability of success $p$. With a Poisson distribution, you essentially have infinite attempts, with infinitesimal chance of success. That is, given a Binomial distribution with some $n,p$, if you let $n\rightarrow\infty$ and $p\rightarrow0$ in such a way that $np\rightarrow\lambda$, then that distribution approaches a Poisson distribution with parameter $\lambda$.
Because of this limiting effect, Poisson distributions are used to model occurences of events that could happen a very large number of times, but happen rarely. That is, they are used in situations that would be more properly represented by a Binomial distribution with a very large $n$ and small $p$, especially when the exact values of $n$ and $p$ are unknown. (Historically, the number of wrongful criminal convictions in a country)
2$\begingroup$ Is the Poisson Distribution (en.wikipedia.org/wiki/Poisson_distribution) different from Poisson Binomial Distribution (en.wikipedia.org/wiki/Poisson_binomial_distribution)? If yes, how? $\endgroup$– MinuOct 2, 2016 at 16:49
1$\begingroup$ A Bernoulli trial is an experiment with two outcomes with fixed probabilities $p$ and $1-p$. The Binomial distribution provides the probability of getting some number of successes amongst a number of Bernoulli trials that have the same $p$ value. The Poisson Binomial distribution, on the other hand, allows for different values of $p$ for each of the individual Bernoulli trials. The Poisson distribution is something different; in this context, it is primarily relevant as a limiting case of the Binomial distribution. $\endgroup$– VyasJun 12, 2017 at 22:32
The binomial distribution counts discrete occurrences among discrete trials.
The poisson distribution counts discrete occurrences among a continuous domain.
Ideally speaking, the poisson should only be used when success could occur at any point in a domain. Such as, for example, cars on a road over a period of time, or random knots in a string over a length, etc. We are talking about infinitely many infinitesimally small trials, each having at most one success.
In practice, though, the poisson can be used to approximate the binomial under certain conditions, but it is only a rough approximation. Such as using the Normal curve in place of a Binomial under the right conditions.
$\begingroup$ Why is this down voted? $\endgroup$ Sep 12, 2017 at 22:21
2$\begingroup$ Why do any of my answers get down-voted? Happens all the time. $\endgroup$ Nov 23, 2017 at 21:42