Glossary

4.4 Poisson Distribution

3 min read•june 25, 2024

The models rare events occurring in fixed intervals. It's used to calculate probabilities for things like customer arrivals or product defects. The formula uses the average event rate and desired number of occurrences to determine likelihood.

have key characteristics like independent events and constant average rates. The distribution can also estimate binomial probabilities when there are many trials with low success rates. This simplifies calculations for large-scale scenarios with rare occurrences.

Poisson Distribution

Poisson distribution probability calculations

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  • Expresses probability of a given number of events occurring within a fixed interval of time or space (minutes, hours, days, area)
  • calculates the probability: P(X=k)=e−λλkk!P(X = k) = \frac{e^{-\lambda}\lambda^k}{k!}
    • λ\lambda average number of events per interval (expected value)
    • ee mathematical constant (2.71828)
    • kk number of events (non-negative integer)
    • k!k! of kk
  • Calculate probability of a specific number of events occurring within an interval:
    1. Identify average number of events per interval (λ\lambda)
    2. Determine desired number of events (kk)
    3. Plug values into Poisson probability mass function
  • Examples:
    • Probability of 3 customers arriving in a 10-minute interval with an average of 2 customers per 10 minutes
    • Probability of 5 defects occurring in a batch of 1000 items with an average defect rate of 0.4%

Key characteristics of Poisson experiments

  • Events occur independently of each other (one event does not affect the probability of another)
  • Average rate of occurrence remains constant over the interval (consistent average number of events per unit of time or space)
  • Two events cannot occur at exactly the same instant (events are discrete)
  • Appropriate when:
    • Number of possible occurrences is large (many potential events)
    • Probability of an event occurring is small (rare events)
    • Events are independent of each other (no influence between events)
  • Mean and variance of a Poisson distribution equal λ\lambda (expected value)
  • Examples:
    • Number of phone calls received by a call center per hour
    • Number of accidents at a busy intersection per day
  • Follows the , which states that the total number of events will follow a Poisson distribution if there are many opportunities for an event to occur, but the probability of occurrence for each opportunity is small

Poisson as binomial distribution estimator

  • Approximates binomial distribution under certain conditions:
    • Number of trials (nn) is large (many repetitions)
    • Probability of success (pp) is small (rare successes)
    • Product of nn and pp is a constant (λ\lambda) (consistent average number of successes)
  • When conditions met, Poisson distribution estimates binomial distribution:
    • λ=np\lambda = np, nn number of trials, pp probability of success
    • Poisson probability mass function approximates binomial probability mass function
  • Advantages of Poisson approximation:
    • Simplifies calculations for large nn and small pp (easier computation)
    • Requires only one parameter (λ\lambda) instead of two (nn and pp) (less information needed)
  • Examples:
    • Approximating the probability of 2 defective items in a large batch of 5000 with a 0.1% defect rate
    • Estimating the probability of 4 successful sales calls out of 200 with a 1.5% success rate
  • : A process where events occur continuously and independently at a constant average rate
  • : A process where the rate of occurrence varies over time or space
  • : Describes the rate of occurrence in a non-homogeneous Poisson process
  • Cumulative distribution function (CDF): Gives the probability that the number of events is less than or equal to a specific value

Key Terms to Review (25)

Binomial Distribution Estimator: The binomial distribution estimator is a statistical tool used to estimate the parameters of a binomial distribution, which is a probability distribution that describes the number of successes in a fixed number of independent Bernoulli trials. It is particularly useful in situations where the outcome of an event can be classified as either a success or a failure, and the trials are independent and have a constant probability of success.
Count Data: Count data refers to data that represents the number of occurrences or the count of a particular event or characteristic within a given time frame or observation period. It is a type of discrete data that can only take on non-negative integer values.
Discrete Probability: Discrete probability refers to the likelihood or chance of an event occurring when the possible outcomes are distinct, separate, and countable. It is concerned with the probabilities of discrete random variables, which can only take on specific, individual values rather than a continuous range of values.
Exponential distribution: The exponential distribution is a continuous probability distribution used to model the time between independent events that occur at a constant average rate. It is characterized by its parameter λ (lambda), which is the rate parameter.
Exponential Distribution: The exponential distribution is a continuous probability distribution that describes the time between events in a Poisson process. It is commonly used to model the waiting time between independent events that occur at a constant average rate.
Factorial: The factorial of a non-negative integer n, denoted as n!, is the product of all positive integers less than or equal to n. It is a fundamental concept in probability and combinatorics that is particularly relevant in the context of the Hypergeometric and Poisson distributions.
Goodness-of-fit test: A goodness-of-fit test determines if a sample data matches a population with a specific distribution. It assesses the discrepancy between observed and expected frequencies.
Goodness-of-Fit Test: The goodness-of-fit test is a statistical test used to determine whether a set of observed data follows a particular probability distribution. It evaluates the discrepancy between the observed and expected frequencies to assess how well the data fits the hypothesized distribution.
Homogeneous Poisson Process: A homogeneous Poisson process is a type of stochastic process that models the occurrence of random events over time. It is characterized by a constant rate of event occurrence, meaning the probability of an event happening in a given time interval is independent of the time elapsed since the last event.
Independence of Events: Independence of events is a fundamental concept in probability theory and statistics, where the occurrence of one event does not influence or depend on the occurrence of another event. This means that the probability of one event happening is not affected by whether another event has occurred or not.
Intensity Function: The intensity function, also known as the rate function or hazard function, is a fundamental concept in the Poisson distribution. It represents the instantaneous rate or likelihood of an event occurring at a given time or location, providing insight into the underlying process generating the observed data.
Lambda: Lambda, in the context of the Poisson distribution, represents the average or expected number of events occurring in a given interval of time or space. It is a fundamental parameter that characterizes the Poisson distribution and determines the probability of observing a specific number of events within that interval.
Law of Rare Events: The law of rare events, also known as the Poisson distribution, describes the probability of a given number of events occurring in a fixed interval of time or space, when these events happen with a known average rate and independently of the time since the last event. This law is particularly useful in modeling and analyzing phenomena where the occurrences are infrequent and random.
Maximum Likelihood Estimation: Maximum likelihood estimation (MLE) is a statistical method used to estimate the parameters of a probability distribution by finding the parameter values that maximize the likelihood of the observed data. It is a fundamental technique in statistical inference that is widely used across various fields, including Poisson distribution and exponential distribution analysis.
Memoryless Property: The memoryless property, also known as the Markov property, is a characteristic of certain probability distributions where the future state of a process depends only on the current state and not on the past states. This property is particularly relevant in the context of the Geometric, Poisson, and Exponential distributions, as it simplifies the analysis and modeling of these probability processes.
Non-Homogeneous Poisson Process: A non-homogeneous Poisson process is a type of stochastic process that models the occurrence of events over time, where the rate of event occurrence can vary depending on the time. This is in contrast to a homogeneous Poisson process, where the rate of event occurrence is constant over time.
Poisson Distribution: The Poisson distribution is a discrete probability distribution that models the number of events occurring in a fixed interval of time or space, given that these events occur with a known average rate and independently of the time since the last event. It is commonly used in various fields, including business, engineering, and the natural sciences, to analyze and predict rare or random events.
Poisson Experiments: Poisson experiments are a type of probability model used to describe the occurrence of discrete, random events over a fixed interval of time or space. They are particularly useful for analyzing rare events or events with a constant rate of occurrence.
Poisson probability distribution: Poisson probability distribution describes the probability of a given number of events occurring within a fixed interval of time or space, assuming these events happen with a known constant mean rate and independently of the time since the last event. It is often used for modeling rare events.
Poisson Probability Formula: The Poisson probability formula is a mathematical equation used to calculate the probability of a specific number of events occurring within a given time period or space, assuming the events occur at a constant average rate and independently of the time since the last event.
Poisson Probability Mass Function: The Poisson probability mass function is a mathematical formula used to calculate the probability of a discrete random variable, such as the number of events occurring within a specific time interval or space, given the average rate at which those events occur. It is a fundamental concept in the Poisson distribution, which is used to model various real-world phenomena that involve the occurrence of random, independent events.
Queueing Theory: Queueing theory is a branch of mathematics that studies the behavior of queues or waiting lines. It provides a framework for analyzing and predicting the performance of systems where customers or tasks arrive, wait in line if necessary, and then are served. Queueing theory is widely applied in various fields, including operations management, computer science, and telecommunications.
Rare Event Modeling: Rare Event Modeling is a statistical approach used to analyze and predict the occurrence of infrequent or unusual events that have a low probability of happening. This type of modeling is particularly relevant in the context of the Poisson distribution, which is commonly used to model the number of rare events occurring within a specific time frame or space.
Rate Parameter: The rate parameter is a fundamental concept in probability theory and statistics that describes the frequency or intensity of a random event or process occurring over time or space. It is a crucial parameter in understanding and modeling various probability distributions, particularly the Poisson distribution and the exponential distribution.
Siméon Denis Poisson: Siméon Denis Poisson was a renowned 19th century French mathematician who made significant contributions to the field of probability theory. His work on the Poisson distribution, a probability distribution that describes the likelihood of a given number of events occurring in a fixed interval of time or space, has become a fundamental concept in statistics and various scientific disciplines.
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