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Introduction to Poisson Distribution
Given the average rate of occurrence, the Poisson distribution is a probability model used to forecast the number of uncommon, discrete events that will occur within a specific length of time or space. It is assumed that events occur independently and at a constant rate. The distribution is widely used in many domains, including queuing theory, insurance, and epidemiology, to analyse and comprehend the possibility of witnessing certain event counts in actual circumstances.
Definition of Poisson Distribution
The Poisson distribution is a probability model that predicts the number of uncommon, discrete events that occur during a particular region of time or space given the average rate of occurrence. It presupposes that events occur independently and at a constant rate. The distribution is widely used in many domains, including queuing theory, insurance, and epidemiology, to analyse and comprehend the chance of witnessing distinct event counts in real-world circumstances.
Poisson Distribution formula:
The probability mass function (PMF) of the Poisson distribution is given by the formula:
- P(x = k) = e-λ λk / k!
- Where: P(X = k) is the probability of observing exactly k events,
- e is the base of the natural logarithm (approximately equal to 2.71828),
- λ is the average rate of events (mean number of events) in the given interval,
- k is the number of events we want to calculate the probability for,
- k! is the factorial of k.
This formula allows us to calculate the probability of observing a specific number of events (k) occurring in a fixed interval, based on the average rate of occurrence (λ).
Poisson Distribution table
- A Poisson distribution table is a tabular representation of probabilities that simplifies the process of seeking up the probabilities in the Poisson distribution for different values of k (number of occurrences). Typically, the table provides the probabilities for various values of k and a given value of (the rate parameter).
- Because calculating Poisson probabilities using the formula for each value of k takes time, these tables serve as a fast reference for typical values of and k. The probabilities in the table are obtained from the Poisson distribution formula: P(x = k) = e-λ λk / k!, where e is the base of the natural logarithm.
Poisson Distribution Mean and variance
In a Poisson distribution, the mean (μ) and variance (σ^2) are both equal to the rate parameter (λ), which represents the average rate of occurrence of events in a fixed interval of time or space. Mathematically:
Mean (μ) = λ
Variance (σ^2) = λ
This unique property of the Poisson distribution makes it particularly useful for modelling situations where rare events occur at a constant rate. The mean represents the average number of events expected in the interval, and the variance indicates the dispersion or spread of the distribution around the mean.
Poisson Distribution expected value
The expected value of a Poisson distribution is equal to its rate parameter (λ), which represents the average rate of occurrence of events in a fixed interval of time or space. Mathematically, the expected value, also known as the mean (μ), is given by E(X) = μ = λ. It signifies the average number of discrete events expected to happen in the interval. For instance, if the rate parameter is λ = 3, then the expected value is also 3, implying an average of three events occurring in that specific time or space interval.
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Solved examples in Poisson Distribution
Example 1: Suppose the average number of cars passing through a particular intersection in a 1-minute interval is 5. What is the probability of exactly 3 cars passing through the intersection in the next 1-minute period?
Solution:
Given: λ (average rate) = 5, k (number of events) = 3
Using the Poisson distribution formula:
P(X = k) = (e^(-λ) * λ^k) / k!
P(X = 3) = (e^(-5) * 5^3) / 3!
P(X = 3) = (e^(-5) * 125) / 6
P(X = 3) ≈ 0.14037
So, the probability of exactly 3 cars passing through the intersection in the next 1-minute period is approximately 0.14037 or 14.04%.
Example 2:In a call center, the average number of customer complaints received in a day is 10. What is the probability of receiving more than 15 complaints in a day?
Solution:
Given: λ (average rate) = 10, k (number of events) > 15
To find the probability of more than 15 complaints, we need to sum the probabilities for k = 16, 17, 18, and so on, up to infinity (since Poisson distribution extends to infinity).
P(X > 15) = 1 – P(X ≤ 15)
Using the Poisson distribution formula to calculate the individual probabilities for k = 0 to 15, and then subtracting from 1, we get:
P(X > 15) ≈ 1 – 0.99993 ≈ 0.00007
So, the probability of receiving more than 15 complaints in a day is approximately 0.00007 or 0.007%. This indicates that such an event is extremely rare.
Frequently asked questions about Poisson Distribution
What are the four conditions for Poisson distribution?
Independent Events: The events that are being tallied must take place independently of one another. In other words, the occurrence of one event should have no bearing on the occurrence of another.
Constant Probability: The likelihood of an event occurring within a certain time interval or region must be constant. This means that the rate at which events occur should be constant throughout time and place.
Discretion: The events being tallied must be discrete, which means they can only have whole integer values (0, 1, 2, 3,...).
Rare Events: The likelihood of more than one event occurring in a very short time interval or in a very compact location should be minimal. In other words, the events being tallied should be uncommon.
What is Poisson Distribution in simple words?
Given the average rate of occurrence, the Poisson distribution is a probability model used to forecast the number of uncommon, discrete events that occur within a specific length of time or space. It is assumed that events occur independently and at a constant rate.
What is Poisson Distribution mean formula?
The mean or expected value (μ) of the Poisson distribution is given by the formula: μ = λ
Where: μ represents the mean or average number of events in the given interval, λ is the rate parameter, which denotes the average rate of occurrence of the events within that interval.
In simple terms, the mean of the Poisson distribution is equal to the average number of events that are expected to happen in a fixed interval of time or space, based on the given rate of occurrence (λ)
What are the uses of Poisson Distribution?
Because of its aptitude for modelling unusual occurrences, the Poisson distribution has various applications in a wide range of domains. It is often used in queueing theory to analyse client arrival rates, telecommunication systems to investigate phone call arrivals, and risk management to simulate unusual calamities or insurance claims. It aids in the estimation of product flaws in production and quality control. Epidemiologists use it to investigate disease outbreak patterns, while environmental scientists use it to investigate unusual events such as earthquakes. The Poisson distribution is used in finance to simulate market event frequencies, and in web traffic monitoring to estimate website visits. Its adaptability makes it useful in healthcare, sports, and a variety of other fields that deal with discrete, rare occurrences.
What are three properties of Poisson Distribution?
The Poisson distribution has three essential properties: Independence: Events must occur independently of one another, which means that the occurrence of one event has no bearing on the occurrence of another.
Constant Rate: The average rate of occurrence of events that remains constant throughout time or space.
What are two properties of Poisson Distribution?
The Poisson distribution has two properties:
Independence: Events must occur independently of one another, which means that the occurrence of one event has no bearing on the occurrence of another.
Constant Rate: The average rate of occurrence of events that remains constant throughout time or space. This rate parameter (lambda) is the same for any specified time or space interval.
What is Poisson Distribution versus binomial distribution?
The Poisson and binomial distributions are both probability distributions used to simulate the number of occurrences or successes in certain scenarios, but they have distinct properties and uses. Definition:
Poisson Distribution: Models the number of uncommon, discrete events that occur in a particular time or space interval, given the average rate of occurrence (λ).
Binomial Distribution: Models the number of successes in a set number of independent Bernoulli trials with the same success probability (p). The number of trials:
There is no idea of a set number of trials in the Poisson Distribution; it focuses on occurrences happening in a continuous or large number of intervals. Binomial Distribution: A fixed number of independent trials (n) is used.
What is Lambda in Poisson Distribution
In the context of the Poisson distribution, lambda (λ) is a parameter that represents the average rate of occurrence of events within a fixed interval of time or space. It is a crucial parameter in the Poisson distribution formula and determines the shape and characteristics of the distribution. Specifically, lambda (λ) denotes the mean or expected number of events that are expected to occur in the given interval
What are the parameters of Poisson Distribution?
The Poisson distribution has two main parameters:
Rate Parameter (λ): This parameter is denoted by λ (lambda) and represents the average rate of occurrence of events within a fixed interval of time or space. It defines the mean or expected number of events in the given interval.
For example, if λ = 3, it means that, on average, three events are expected to occur in that specific time period. Number of Events (k): This is the variable for which the probability is calculated. It represents the number of rare, discrete events that are observed in the given interval. With these two parameters, the Poisson distribution allows us to calculate the probability of observing a specific number of events (k) occurring in a given interval, based on the average rate of occurrence (λ).
The probability formula is given as P(x = k) = e-λ λk / k!, where e is the base of the natural logarithm.
