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Poisson Distribution: Definition, Formula, and Examples

Poisson Distribution

Definition: 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 if these events occur with a known constant mean rate and independently of the time since the last event. For example, it can model the number of phone calls received by a call center in an hour or the number of cars passing through a toll booth in a day.

Formula:

P(X = k) = \frac{e^{-\lambda}\lambda^k}{k!}

Where:

$k$ is the number of occurrences (usually an integer)
$\lambda$ is the average number of events in the interval
$e$ is Euler's number $(e \approx 2.71828)$

Examples:

Suppose a call center receives an average of 5 calls per hour. Let's calculate various probabilities:

Probability of receiving exactly 3 calls in an hour: $P(X = 3) = \frac{e^{-5}5^3}{3!} \approx 0.1404$
Probability of receiving at most 2 calls in an hour: $P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2) \approx 0.1247$
Probability of receiving more than 6 calls in an hour: $P(X > 6) = 1 - P(X \leq 6) \approx 0.2851$

Properties of Poisson Distribution

Mean: $E(X) = \lambda$
Variance: $Var(X) = \lambda$
Standard Deviation: $SD(X) = \sqrt{\lambda}$
Poisson as an approximation of the Binomial distribution: If the number of trials $n$ is large, the probability of success $p$ is small, and the product $np$ is moderate, the Binomial distribution can be approximated by the Poisson distribution with $\lambda = np$ .

Applications

The Poisson distribution is widely used in various fields:

1. Quality Control in Manufacturing

The Poisson distribution is frequently used to model the occurrence of defects in manufacturing processes. Since defects are typically rare events, it fits well with the distribution’s assumption of modeling low-probability events over a continuous interval. For example, if a manufacturer produces a large number of units and expects a small number of defective products in a batch, the Poisson distribution can predict the likelihood of observing a certain number of defects per batch. This helps in setting acceptable quality limits, identifying faulty processes, and determining the required sample sizes for quality control.

2. Traffic Flow Analysis

In traffic engineering, the Poisson distribution models the arrival of vehicles at a specific point on a road over time. The key assumption is that car arrivals are independent events, and the average arrival rate (vehicles per unit of time) is constant. For example, it can be used to estimate the probability that exactly 10 cars will arrive at a traffic light in a 5-minute period. This kind of modeling is useful for designing traffic signals, optimizing road layouts, and improving traffic management systems. It is also applicable in areas like public transportation to predict passenger arrivals at stations.

3. Queueing Theory

In queueing systems, the Poisson distribution helps model the arrival of customers, calls, or tasks at service facilities such as bank counters, call centers, or network servers. The assumption is that arrivals occur randomly and independently over time, with a constant average rate. For instance, a call center may use the Poisson distribution to model the number of incoming customer calls during an hour. This helps determine optimal staffing levels, predict wait times, and manage resources efficiently to ensure smooth operation and avoid long queues.

4. Insurance and Finance

In the insurance and finance sectors, the Poisson distribution models the occurrence of rare but significant events, such as insurance claims or loan defaults. For instance, an insurance company might use it to estimate the number of claims expected in a given period based on historical data. This helps insurers set premiums and reserve funds accordingly. In finance, it can also model the occurrence of credit defaults or other rare financial events over a specified time frame, assisting in risk management and portfolio analysis.

5. Biology and Medicine

In biology and medicine, the Poisson distribution is applied to model rare biological events, such as mutation rates in DNA sequences or the occurrence of a specific disease in a population over time. For example, geneticists may use it to predict the number of mutations that will occur in a certain length of DNA during cell replication. Similarly, epidemiologists can use it to model the spread of infectious diseases, particularly in the early stages of an outbreak, where the number of cases in a given area is small and can be treated as a rare event. This information is crucial in designing interventions and public health strategies.

How to calculate Poisson distribution in R?

1# Set parameter
2lambda <- 5  # average rate
3
4# Calculate P(X = 3)
5prob_equal_3 <- dpois(3, lambda)
6print(paste("P(X = 3):", prob_equal_3))
7
8# Calculate P(X <= 2)
9prob_less_equal_2 <- ppois(2, lambda)
10print(paste("P(X <= 2):", prob_less_equal_2))
11
12# Calculate P(X > 6)
13prob_greater_6 <- 1 - ppois(6, lambda)
14print(paste("P(X > 6):", prob_greater_6))
15
16# Calculate mean and variance
17mean <- lambda
18variance <- lambda
19print(paste("Mean:", mean))
20print(paste("Variance:", variance))

How to calculate Poisson distribution in Python?

Python

1import scipy.stats as stats
2
3# Set parameter
4lambda_ = 5  # average rate
5
6# Calculate P(X = 3)
7prob_equal_3 = stats.poisson.pmf(3, lambda_)
8print(f"P(X = 3): {prob_equal_3:.4f}")
9
10# Calculate P(X <= 2)
11prob_less_equal_2 = stats.poisson.cdf(2, lambda_)
12print(f"P(X <= 2): {prob_less_equal_2:.4f}")
13
14# Calculate P(X > 6)
15prob_greater_6 = 1 - stats.poisson.cdf(6, lambda_)
16print(f"P(X > 6): {prob_greater_6:.4f}")
17
18# Calculate mean and variance
19mean = lambda_
20variance = lambda_
21print(f"Mean: {mean}")
22print(f"Variance: {variance}")