1. Meaning of a Poisson Distribution
A Poisson distribution is used to model the probability of a certain number of events happening in a fixed interval of time or space when the events occur rarely and independently.
It helps when events happen randomly but with a known average rate.
1.1. Examples
- Number of phone calls received by a help desk in 1 minute.
- Number of printing errors per page of a book.
- Number of cars passing a point in 30 seconds.
2. When Poisson Distribution is Used
A Poisson distribution is suitable when the following conditions are met:
2.1. Events Are Rare
The events occur infrequently compared to the time or space interval.
2.2. Events Occur Independently
One event happening does not affect the chance of another event happening.
2.3. Constant Average Rate
The events happen with a fixed average rate, denoted by \( \lambda \) (lambda).
2.4. Events Occur One at a Time
Two or more events do not occur at exactly the same moment.
3. Poisson Probability Formula
If X is the number of events in a fixed interval, then the probability that exactly k events occur is:
\( P(X = k) = \dfrac{\lambda^k e^{-\lambda}}{k!} \)
Here:
- \( k \) = number of events
- \( \lambda \) = average rate of events
- \( e \approx 2.718 \)
3.1. Example
If a call centre receives an average of 3 calls per minute, what is the probability of receiving exactly 2 calls in a minute?
Here \( \lambda = 3 \) and \( k = 2 \).
\( P(X = 2) = \dfrac{3^2 e^{-3}}{2!} = \dfrac{9e^{-3}}{2} \)
4. Mean and Variance
A Poisson distribution has equal mean and variance, both given by:
\( \mu = \lambda, \quad \sigma^2 = \lambda \)
4.1. Example
If the average number of misprints per page is 0.8, then:
- Mean = 0.8
- Variance = 0.8
5. More Examples
Situations where Poisson distribution applies:
5.1. Accidents or Failures
Number of accidents happening in a day or number of machine breakdowns in a week.
5.2. Requests or Arrivals
Number of customers arriving at a shop in 10 minutes.
5.3. Counts Over Space
Number of bacteria in a square centimetre of a slide.