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. The events are assumed to occur with a known constant rate and independently of the time since the last event.
Definition
The Poisson distribution is characterized by its parameter λ (lambda), which is the average number of occurrences (mean) within the interval. The probability mass function (PMF) of the Poisson distribution is given by:
\[ P(X = k) = \frac{λ^k e^{-λ}}{k!} \]
Where:
- \( P(X = k) \) is the probability that there are \( k \) events in the interval.
- \( λ \) is the average number of events per interval.
- \( k \) is the number of events.
- \( e \) is Euler’s number (approximately equal to 2.71828).
Examples
Example 1: Number of Emails Received
If a person receives an average of 5 emails per hour, the number of emails received in any given hour follows a Poisson distribution with \( λ = 5 \).
Example 2: Calls at a Call Center
Suppose a call center receives an average of 10 calls per hour. The number of calls received in one hour follows a Poisson distribution with \( λ = 10 \).
Frequently Asked Questions (FAQs)
What is the difference between the Poisson distribution and the binomial distribution?
The Poisson distribution models the number of events in a fixed interval where the events occur independently and with a constant mean rate, while the binomial distribution models the number of successes in a fixed number of independent trials with a constant probability of success.
What assumptions are made when using the Poisson distribution?
The Poisson distribution assumes that the events occur independently, the average rate \( λ \) is constant, and each small interval is such that the probability of more than one event occurring is negligible.
How is the Poisson distribution used in real life?
The Poisson distribution is used in various fields such as telecommunications to model call arrivals, in traffic engineering to model the number of cars passing through an intersection, and in biology to model the number of mutations in a given DNA segment.
Related Terms
Exponential Distribution
The exponential distribution describes the time between events in a Poisson process. It is continuous and complements the Poisson distribution.
Binomial Distribution
The binomial distribution is a discrete probability distribution of the number of successes in a sequence of independent yes/no experiments, each of which yields success with a given probability.
Probability Mass Function
The probability mass function (PMF) specifies the probability that a discrete random variable is exactly equal to some value.
Online References
Suggested Books for Further Studies
- “Probability and Statistics for Engineers and Scientists” by Ronald E. Walpole, Raymond H. Myers, Sharon L. Myers, and Keying E. Ye.
- “Introduction to Probability Models” by Sheldon M. Ross.
- “Statistical Inference” by Casella and Berger.
Fundamentals of Poisson Distribution: Statistics Basics Quiz
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