The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space.
These events must occur independently and at a constant average rate.
Formula
Applications of Poisson's Distribution
The Poisson distribution is utilized across different fields to model the frequency of events within a defined interval.
Some examples include:
The number of calls a call center receives per hour.
The frequency of decay events from a radioactive source over a given time period.
The number of emails received daily.
Properties
Discrete Distribution: Applicable to discrete (countable) events.
Parameter: Characterized by a single parameter, λ, which represents the average rate.
Mean and Variance: Both the mean and variance are equal to λ.
Pharmaceutical Example
Suppose a pharmaceutical company wants to model the number of side effects occurring in patients taking a new drug.
If the average number of side effects per day is 2, they can use the Poisson distribution to determine the probability of observing a certain number of side effects.
For example, to find the probability of exactly 3 side effects occurring in a day (λ = 2, k = 3k):
Thus, the probability of observing exactly 3 side effects in a day is approximately 0.180 or 18%.
This example illustrates how the Poisson distribution can be applied to model the occurrence of rare events in pharmaceutical studies, helping researchers and healthcare professionals to understand and manage the risks associated with new treatments.