# Poisson Distribution

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It is a discrete distribution function describing the probability that an event will occur a certain number of times in a fixed time (or space) interval.
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It is used to model count-based data, like the number of emails arriving in your mailbox in one hour or the number of customers walking into a shop in one day, for instance. It can be used to predict how many times an event might occur in a given time period. 

* The number (k) of hits on a website in one hour with an average hit rate of 6 hits per hour is poisson distributed.
* Insurance companies to conduct risk analysis (eg. predict the number of car crash accidents within a predefined time span) to decide car insurance pricing.

**The Poisson distribution is parametrized by the expected number of events λ** (pronounced “lambda”) in a time or space window. The distribution is a function that takes the **number of occurrences of the event as input** (the integer called k in the next formula) and **outputs the corresponding probability** (the probability that there are k events occurring).

### The Probability Mass Function

$$
P(k;\lambda) = \frac{\lambda^k e^{-\lambda}}{k!}
$$

for k = 0, 1, 2, ...\
The formula of P(k; λ) returns the probability of observing k events given the parameter λ which corresponds to the expected (sometimes called average) number of occurrences in that time slot.

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**Poisson Process:** The timing of the next event is completely independent of when the previous event happened.
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### Example

![](https://2552912007-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-LzGBVuquaFNrwdmJna0%2F-MPvP5MxgExMSR3U_V5U%2F-MPvPXBBIFSlAszX1b3B%2FScreen%20Shot%202020-12-31%20at%205.47.06%20PM.png?alt=media\&token=6de2697a-5467-4c39-a269-b93c5e5e9e9c)

![](https://2552912007-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-LzGBVuquaFNrwdmJna0%2F-MPvP5MxgExMSR3U_V5U%2F-MPvPcasJOSspnXfPiY0%2FScreen%20Shot%202020-12-31%20at%205.47.58%20PM.png?alt=media\&token=d9eecc97-991b-4023-a056-f63d4378c877)

![](https://2552912007-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-LzGBVuquaFNrwdmJna0%2F-MPvP5MxgExMSR3U_V5U%2F-MPvPjBS4YXhJDsakFhT%2FScreen%20Shot%202020-12-31%20at%205.48.29%20PM.png?alt=media\&token=a99c0253-1d87-40df-b2e9-0eb18d50ebf1)

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Say we do a Bernoulli trial every minute for an hour, each with a success probability of 0.1. We would do 60 trials, and the number of successes is Binomially distributed, and we would expect to get about 6 successes. This is just like the Poisson story we discussed, where we get on average 6 hits on a website per hour. 

So, the Poisson distribution with arrival rate equal to **np** approximates a Binomial distribution for **n** Bernoulli trials with probability **p** of success (with **n** large and **p** small). Importantly, the Poisson distribution is often simpler to work with because it has only one parameter instead of two for the Binomial distribution.
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{% embed url="<https://www.kdnuggets.com/2020/12/introduction-poisson-distribution-data-science.html>" %}

{% embed url="<https://www.youtube.com/watch?v=m0o-585xwW0>" %}