A Poisson distribution is defined as how many times an event occurs within `x`

fixed interval. Interval can be minutes, hours, days or any unit of time. Not just time - it can be any interval - for example the number of misspellings in a page.

Solution

P(X < ) means probability of less than occurences. Similarly P(X > ) refers to probability of more than occurences.

Probabilities of exactly N occurences from 0 through are shown below.

Examples

See below some practical examples of poisson distribution in real life

**Arrivals at a bus stop:**If buses arrive at a bus stop at an average rate of 3 buses per hour, the Poisson distribution can be used to calculate the probability of a specific number of arrivals in the next hour, such as the probability of exactly 5 buses arriving. λ=3 represents the average rate of arrivals.**Website traffic:**If a website receives an average of 1000 visitors per day, the Poisson distribution can be used to calculate the probability of a certain number of visitors on a specific day, such as the probability of 1200 visitors. λ=1000 represents the average number of visitors.**Defects in a product:**If a factory produces light bulbs and the average number of defects is 2 per batch of 100 bulbs, the Poisson distribution can be used to calculate the probability of a specific number of defects in a batch, such as the probability of exactly 1 defect. λ=2 represents the average number of defects per batch.**Accidents:**If the average number of car accidents at a particular intersection is 0.5 per day, the Poisson distribution can be used to calculate the probability of a specific number of accidents in the next day, such as the probability of no accidents. λ=0.5 represents the average number of accidents per day.**Call centre :**Customer care executive gets three phone calls every 15 mins. What is the probability that he/she would get more than 5 calls in the next 30 minutes? Given the average rate of success is 3 calls for every 15 minutes. Therefore, average rate of success (λ) for our problem statement which is to calculate for 30 minutes would be 3 x (30/15) = 6. Number of occurences (x) is 5. It is also called`Poisson random variable`

.

Important Points

- Average rate of success (λ) must be greater than or equal to 0.
- Poisson random variable must be a whole number, can't be in decimals.
- Poisson random variable can't be less than zero.

Properties of poisson distribution

Poisson Distribution has the following properties -
- Average rate of success must be already known.
- Probability of a single occurence is proportional to the size of the interval.
- Probability of more than one occurence within a very short interval is small.