Difference between Binomial Distribution and Poisson Distribution

Binomial vs. Poisson

Binomial Distribution	Poisson Distribution
Fixed Number of Trials (n) [10 pie throws]	Infinite Number of Trials
Only 2 Possible Outcomes [hit or miss]	Unlimited Number of Outcomes Possible
Probability of Success is Constant (p) [0.4 success rate]	Mean of the Distribution is the Same for All Intervals (mu)
Each Trial is Independent [throw 1 has no effect on throw 2]	Number of Occurrences in Any Given Interval Independent of Others
Predicts Number of Successes within a Set Number of Trials	Predicts Number of Occurrences per Unit Time, Space, ...
Can be Used to Test for Independence	Can be Used to Test for Independence

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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$n$, of "attempts," each of which has probability of success p$p$. With a Poisson distribution, you essentially have infinite attempts, with infinitesimal chance of success. That is, given a Binomial distribution with some n,p$n,p$, if you let n→∞$n\to \mathrm{\infty }$ and p→0$p\to 0$ in such a way that np→λ$np\to \lambda$, then that distribution approaches a Poisson distribution with parameter λ$\lambda$. Because of this limiting effect, Poisson distributions are used to model occurences of events thatcould 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$n$ and small p$p$, especially when the exact values of n$n$ and p$p$ are unknown. (Historically, the number of wrongful criminal convictions in a country)