Glossary

10.2 Poisson distribution

2 min read•july 19, 2024

The Poisson process models rare events in continuous domains like time or space. It assumes events occur randomly, independently, and at a constant average rate. The process is memoryless, has stationary increments, and follows the orderliness property.

The , derived from the binomial distribution, has a single parameter λ representing the average number of events in a fixed interval. It's useful for calculating probabilities of exact, at least, or at most k events occurring.

Poisson Process and Distribution

Poisson process and assumptions

Top images from around the web for Poisson process and assumptions

  • Probability distribution - wikidoc View original

    Is this image relevant?

  • Poisson distribution - Wikipedia View original

    Is this image relevant?

  • Probability distribution - wikidoc View original

    Is this image relevant?

1 of 3

Top images from around the web for Poisson process and assumptions

  • Probability distribution - wikidoc View original

    Is this image relevant?

  • Poisson distribution - Wikipedia View original

    Is this image relevant?

  • Probability distribution - wikidoc View original

    Is this image relevant?

1 of 3
  • Models occurrence of rare events over continuous domain (time, space, volume)
    • Events occur randomly and independently
    • Average rate of occurrence remains constant
    • Probability of event in small interval proportional to interval length
  • Key assumptions:
    • Memoryless property: Number of events in non-overlapping intervals are independent
    • Stationary increments: Probability distribution depends only on interval length, not location
    • Orderliness: Probability of multiple events simultaneously in small interval negligible compared to single event

Poisson formula derivation

  • Consider binomial distribution with parameters nn (trials) and pp (success probability)
  • Let λ=np\lambda = np be average number of successes in nn trials
  • As n→∞n \to \infty and p→0p \to 0 while keeping λ\lambda constant, binomial distribution converges to Poisson distribution
  • Poisson distribution has single parameter λ\lambda, representing average number of events in fixed interval
  • Probability mass function (PMF) of Poisson distribution:
    • P(X=k)=e−λλkk!P(X = k) = \frac{e^{-\lambda}\lambda^k}{k!}, where k=0,1,2,…k = 0, 1, 2, \ldots

Poisson probability calculations

  • Calculate probability of exactly kk events in fixed interval using Poisson PMF:
    • P(X=k)=e−λλkk!P(X = k) = \frac{e^{-\lambda}\lambda^k}{k!}
  • Calculate probability of at least kk events by summing probabilities for all values ≥k\geq k:
    • P(X≥k)=∑i=k∞e−λλii!P(X \geq k) = \sum_{i=k}^{\infty} \frac{e^{-\lambda}\lambda^i}{i!}
  • Calculate probability of at most kk events by summing probabilities for all values ≤k\leq k:
    • P(X≤k)=∑i=0ke−λλii!P(X \leq k) = \sum_{i=0}^{k} \frac{e^{-\lambda}\lambda^i}{i!}

Applications and Comparison

Poisson modeling for rare events

  • Suitable for modeling rare events in various fields:
    • Quality control (defects in large batch of products)
    • Biology (mutations in DNA sequence)
    • Telecommunications (phone calls arriving at call center)
    • Insurance (claims filed by policyholders)
  • Ensure key assumptions of Poisson process are satisfied when applying distribution

Binomial vs Poisson distributions

  • Binomial distribution:
    • Models number of successes in fixed number of independent trials
    • Two parameters: nn (number of trials) and pp (success probability in each trial)
    • Trials are discrete and finite
  • Poisson distribution:
    • Models number of rare events in fixed interval (time, space, volume)
    • Single parameter λ\lambda, representing average number of events in interval
    • Events occur continuously and independently over interval
  • Poisson distribution approximates binomial distribution when:
    • Number of trials nn is large
    • Success probability pp is small
    • Product npnp remains constant as λ\lambda

Key Terms to Review (15)

Arrival of Customers: The arrival of customers refers to the process by which individuals or entities enter a service system seeking to obtain a service or product. This concept is crucial in understanding customer flow, which can be modeled using the Poisson distribution, as it helps to analyze and predict the timing and frequency of these arrivals within a specified period, aiding businesses in managing resources effectively.
Central Limit Theorem: The Central Limit Theorem (CLT) states that the distribution of the sum (or average) of a large number of independent and identically distributed random variables approaches a normal distribution, regardless of the original distribution of the variables. This key concept bridges many areas in statistics and probability, establishing that many statistical methods can be applied when sample sizes are sufficiently large.
Confidence Interval: A confidence interval is a range of values that is used to estimate the true value of a population parameter with a certain level of confidence. This statistical tool helps quantify the uncertainty around sample estimates, providing a lower and upper bound within which the true parameter is likely to fall. By expressing results in this way, it facilitates decision-making and risk assessment in various fields, particularly when dealing with distributions like the Poisson distribution and techniques such as Monte Carlo simulations.
Event Rate: Event rate is a measure that quantifies the frequency at which specific events occur in a given time frame or space, often expressed as the average number of events per interval. It serves as a critical parameter in both the Poisson distribution and Poisson processes, helping to model and predict the likelihood of events happening over time or within a specified area. Understanding event rate is fundamental for analyzing scenarios where events occur independently and sporadically.
Expected Value: Expected value is a fundamental concept in probability that quantifies the average outcome of a random variable over numerous trials. It serves as a way to anticipate the long-term results of random processes and is crucial for decision-making in uncertain environments. This concept is deeply connected to randomness, random variables, and probability distributions, allowing us to calculate meaningful metrics such as averages, risks, and expected gains or losses.
Hypothesis Testing: Hypothesis testing is a statistical method used to make decisions about population parameters based on sample data. It involves formulating a null hypothesis and an alternative hypothesis, then using sample statistics to determine whether there is enough evidence to reject the null hypothesis in favor of the alternative. This process connects to various statistical concepts and distributions, allowing for applications in different fields.
Lambda: Lambda is a parameter in the Poisson distribution that represents the average number of events occurring in a fixed interval of time or space. It plays a crucial role in determining the shape and characteristics of the Poisson probability distribution, indicating how frequently events happen on average. A higher value of lambda indicates that events are more likely to occur in the given interval.
Law of Rare Events: The law of rare events states that in a large enough sample, the occurrence of rare events can be modeled using a Poisson distribution. This principle connects the likelihood of observing infrequent occurrences to a predictable mathematical framework, allowing for accurate predictions even when events are uncommon. As events become increasingly rare, this law becomes particularly useful, demonstrating how such occurrences can be understood through statistical analysis.
Mean: The mean, often referred to as the average, is a measure of central tendency that quantifies the expected value of a random variable. It represents the balancing point of a probability distribution, providing insight into the typical outcome one can expect from a set of data or a probability distribution. The concept of the mean is essential in understanding various statistical properties and distributions, as it lays the foundation for further analysis and interpretation.
Number of defects in manufacturing: The number of defects in manufacturing refers to the count of items produced that fail to meet quality standards during the production process. This measurement is critical for assessing the efficiency of manufacturing processes and the overall quality of products, allowing companies to identify areas for improvement and reduce waste. By understanding the patterns and frequency of defects, manufacturers can implement better quality control measures and improve their operational effectiveness.
Poisson distribution: The Poisson distribution is a probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space, provided that these events occur with a known constant mean rate and independently of the time since the last event. This distribution connects to several concepts, including randomness and discrete random variables, which can help quantify uncertainties in various applications, such as queuing systems and random signals.
Poisson Probability Mass Function: The equation $$p(x=k) = \frac{\lambda^k e^{-\lambda}}{k!}$$ represents the probability of observing exactly k events in a fixed interval when these events happen with a known average rate, λ, and independently of the time since the last event. This formula connects to several key aspects such as the nature of rare events, the concept of independence, and the applications in various fields like engineering and natural sciences where such random occurrences are modeled. It is essential in determining how likely different counts of occurrences are based on the average rate.
Queueing Theory: Queueing theory is the mathematical study of waiting lines or queues, focusing on the behavior of queues in various contexts. It examines how entities arrive, wait, and are served, which is essential for optimizing systems in fields like telecommunications, manufacturing, and service industries. Understanding queueing theory helps to model and analyze systems where demand exceeds capacity, making it crucial for effective resource allocation and operational efficiency.
Random Events: Random events are occurrences that cannot be predicted with certainty due to the influence of chance. In probability theory, these events are fundamental as they form the basis for understanding various distributions and modeling real-world phenomena. Each random event can have one or more outcomes, which can be described using different probabilistic models, such as the Poisson distribution, to analyze and make predictions about the likelihood of these events happening within a given timeframe or space.
Reliability Engineering: Reliability engineering is a field of engineering that focuses on ensuring a system's performance and dependability over its intended lifespan. It involves the use of statistical methods and probability theory to predict failures and improve system reliability, often by analyzing various factors such as random variables and distributions. The aim is to minimize risks and enhance safety in systems, which connects to various aspects of uncertainty and variability in performance.
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
Back
Practice QuizGlossary
Practice QuizGlossary
Next guide