How to Calculate Poisson Distribution PMF

The Poisson distribution is a discrete distribution which describes the number of events that occur in a fixed time interval. The Poisson distribution requires only one parameter λ and is bounded by 0 and *∞.* Furthermore, the Poisson distribution assumes that events take place at a constant rate and that events are independent, meaning that any one event does not affect the probability of a subsequent event taking place.
If a small store gains 10 customers on average every hour. What is the probability that it will get exactly 15 customers in a given hour? To answer this question we need to calculate the probability mass function or PMF.
The formula for the Poisson Distribution PMF, that is to say the probability that EXACTLY a given number of events happen can be written as P(x; λ) = (e^-λ * λ^x) / x!, where:
- x = the number of events we’re interested in, in this case 15.
- λ = the average number of events in a given time period, in this case 10.
- e = Euler’s number, which is a constant equal to 2.71828.
- ! = factorial, which is the number X*(X-1)*X-2… until *1 is reached.
If we now plug in our numbers we get (2.71828^-10 * 10^15) / 15! = 0.0347 → 3.47%
Other real world applications of the Poisson distribution includes:
- Counting the number of defects of a finished product.
- Counting the number of deaths in a country by any disease or natural calamity.
- Counting the number of infected plants in the field.
- Counting the number of bacteria in the organisms or the radioactive decay in atoms.
- Calculating the waiting time between events.