Poisson Distribution

Business / By / November 21, 2023

Poisson Distribution is a discrete probability distribution used to model the number of events occurring within a fixed interval with a constant mean rate (λ). It finds applications in queueing theory, insurance risk assessment, and particle physics. However, it may not be suitable for rare events and relies on the assumption of event independence.

The Enlightened Accountant by Gennaro Cuofano – FourWeekMBADownload

Poisson Distribution is a fundamental probability distribution used to model the number of events occurring within a fixed interval of time or space when these events happen with a known constant mean rate and are statistically independent of the time since the last event.

It is named after the French mathematician Siméon Denis Poisson, who made significant contributions to the field of probability theory. Poisson Distribution finds extensive applications in various fields, including statistics, queuing theory, insurance, and particle physics.

Characteristics of Poisson Distribution:

  • Mean and Variance Equality: One of the defining characteristics of Poisson Distribution is that its mean (λ, lambda) is equal to its variance (λ). This equality simplifies the distribution’s parameters and makes it particularly useful for modeling certain types of events.
  • Discrete Distribution: Poisson Distribution is a discrete probability distribution, meaning it deals with countable and distinct events. It is suitable for modeling events that occur at specific points in time or space.

Probability Mass Function (PMF) Formula:

The probability mass function (PMF) of Poisson Distribution is given by the following formula:

P(X = k) = (e^-λ * λ^k) / k!

Where:

  • P(X = k) represents the probability of observing k events in the given interval.
  • λ (lambda) is the average rate at which events occur in the interval.
  • e is Euler’s number, approximately equal to 2.71828.
  • k is the number of events we want to calculate the probability for.
  • k! represents the factorial of k.

Key Characteristics of the Poisson Distribution

The Poisson distribution exhibits several key characteristics:

  1. Rare Events: It is used to model rare events, where the probability of more than one event occurring in a very short time or space interval is negligible.
  2. Discreteness: The Poisson distribution deals with discrete random variables (non-negative integers) and is not applicable to continuous quantities.
  3. Memorylessness: The Poisson distribution is memoryless, meaning that the probability of future events is independent of past events. For example, if a machine is expected to fail at a constant rate, the time until the next failure follows a Poisson distribution and is not influenced by the time since the last failure.
  4. Mean and Variance: The mean (μ) and variance (2σ2) of a Poisson distribution are both equal to λ, which represents the average rate of events. Thus, =2=μ=σ2=λ.
  5. Shape: The shape of the Poisson distribution depends on the value of λ. When λ is small, the distribution is skewed to the right. As λ increases, the distribution becomes more symmetric and approaches a normal distribution.

Applications of the Poisson Distribution

The Poisson distribution finds applications in various fields due to its ability to model rare events accurately:

1. Queuing Theory

In queuing theory, the Poisson distribution is used to model the arrival rate of customers at service points, such as banks, call centers, and healthcare facilities. It helps optimize staffing and resource allocation.

2. Insurance

In insurance, the Poisson distribution is employed to model the number of insurance claims filed by policyholders. This information is crucial for calculating premiums and assessing risk.

3. Epidemiology

Epidemiologists use the Poisson distribution to model the occurrence of disease outbreaks or rare events like epidemics. It helps predict the spread of diseases and allocate resources for disease control.

4. Manufacturing and Quality Control

Manufacturers use the Poisson distribution to analyze defect rates and failure rates in production processes. Quality control methods often rely on Poisson distributions to monitor and improve product quality.

5. Environmental Sciences

Environmental scientists may use the Poisson distribution to model rare environmental events, such as earthquakes, hurricanes, or pollutant discharges. It aids in risk assessment and disaster preparedness.

Poisson Distribution vs. Other Distributions

While the Poisson distribution is used for modeling rare events, it is essential to distinguish it from other probability distributions with similar characteristics:

  1. Binomial Distribution: The binomial distribution is used to model the number of successful outcomes in a fixed number of independent Bernoulli trials. It is suitable for events with two possible outcomes, such as success or failure. In contrast, the Poisson distribution models the number of events occurring within a fixed interval, and the events are assumed to be rare and independent.
  2. Exponential Distribution: The exponential distribution is used to model the time between rare events in a continuous time domain. It describes the waiting time until the next event in a Poisson process. The Poisson distribution models the count of events within a fixed interval, while the exponential distribution models the time between events.
  3. Normal Distribution: The normal distribution is suitable for continuous data and is not appropriate for modeling discrete counts of rare events. It is characterized by a bell-shaped curve and is used for continuous variables that follow a normal distribution.

Conclusion

The Poisson distribution is a valuable tool in probability theory and statistics for modeling rare events with discrete counts. Its applications span various fields, including queuing theory, insurance, epidemiology, manufacturing, and environmental sciences. Understanding the key characteristics and applications of the Poisson distribution is essential for making informed decisions and predictions when dealing with rare and random events. Whether analyzing customer arrivals at a store, insurance claims, or disease outbreaks, the Poisson distribution provides a reliable framework for probability modeling and risk assessment.

Real-World Example: Insurance Claims

Suppose an insurance company wants to predict the number of auto insurance claims it is likely to receive in a month based on historical data. Let’s assume that, on average, the company receives 3 claims per month (λ = 3).

Using Poisson Distribution, the company can calculate the probability of observing a specific number of claims in a month. For example, it can determine the probability of receiving exactly 5 claims in a given month:

P(X = 5) = (e^-3 * 3^5) / 5! ≈ 0.1008

This means there is approximately a 10.08% chance of the company receiving exactly 5 claims in the upcoming month.

In summary, Poisson Distribution is a valuable tool for modeling events that occur over time or space when events are independent and occur at a constant average rate.

Its applications range from queueing theory to insurance risk assessment and particle physics.

However, it should be used with caution, especially when dealing with rare events or situations where the assumption of independence does not hold.

Key Highlights of Poisson Distribution:

  • Mean-Variance Equality: In Poisson Distribution, the mean (λ) is equal to the variance, simplifying the distribution’s parameters.
  • Discrete Probability Distribution: Poisson is a discrete distribution that models countable and distinct events.
  • Probability Mass Function (PMF): The PMF calculates the probability of observing a specific number of events within an interval.
  • Applications in Queueing Theory: Poisson Distribution is widely used to analyze queues and optimize systems with applications in call centers, traffic management, and more.
  • Insurance Risk Assessment: Insurance companies employ it to predict and manage the number of insurance claims within a given period.
  • Particle Physics: Poisson Distribution plays a role in particle physics for studying particle decays and interactions.
  • Assumption of Independence: It assumes events are independent, which may not always hold in real-world scenarios.
  • Not Ideal for Rare Events: Poisson is less suitable for modeling rare events where events are infrequent.
  • Real-World Example: Insurance companies use Poisson to predict monthly insurance claims based on historical data and average claim rates.
  • Probability Calculations: Poisson allows calculating the probability of observing specific event counts within a defined interval.
  • Limitations: It has limitations when applied to continuous processes or when events deviate from independence assumptions.
  • Versatility: Despite its limitations, Poisson Distribution remains a versatile tool for modeling events with a constant average rate.

Connected Financial Concepts

Circle of Competence

circle-of-competence
The circle of competence describes a person’s natural competence in an area that matches their skills and abilities. Beyond this imaginary circle are skills and abilities that a person is naturally less competent at. The concept was popularised by Warren Buffett, who argued that investors should only invest in companies they know and understand. However, the circle of competence applies to any topic and indeed any individual.

What is a Moat

moat
Economic or market moats represent the long-term business defensibility. Or how long a business can retain its competitive advantage in the marketplace over the years. Warren Buffet who popularized the term “moat” referred to it as a share of mind, opposite to market share, as such it is the characteristic that all valuable brands have.

Buffet Indicator

buffet-indicator
The Buffet Indicator is a measure of the total value of all publicly-traded stocks in a country divided by that country’s GDP. It’s a measure and ratio to evaluate whether a market is undervalued or overvalued. It’s one of Warren Buffet’s favorite measures as a warning that financial markets might be overvalued and riskier.

Venture Capital

venture-capital
Venture capital is a form of investing skewed toward high-risk bets, that are likely to fail. Therefore venture capitalists look for higher returns. Indeed, venture capital is based on the power law, or the law for which a small number of bets will pay off big time for the larger numbers of low-return or investments that will go to zero. That is the whole premise of venture capital.

Foreign Direct Investment

foreign-direct-investment
Foreign direct investment occurs when an individual or business purchases an interest of 10% or more in a company that operates in a different country. According to the International Monetary Fund (IMF), this percentage implies that the investor can influence or participate in the management of an enterprise. When the interest is less than 10%, on the other hand, the IMF simply defines it as a security that is part of a stock portfolio. Foreign direct investment (FDI), therefore, involves the purchase of an interest in a company by an entity that is located in another country. 

Micro-Investing

micro-investing
Micro-investing is the process of investing small amounts of money regularly. The process of micro-investing involves small and sometimes irregular investments where the individual can set up recurring payments or invest a lump sum as cash becomes available.

Meme Investing

meme-investing
Meme stocks are securities that go viral online and attract the attention of the younger generation of retail investors. Meme investing, therefore, is a bottom-up, community-driven approach to investing that positions itself as the antonym to Wall Street investing. Also, meme investing often looks at attractive opportunities with lower liquidity that might be easier to overtake, thus enabling wide speculation, as “meme investors” often look for disproportionate short-term returns.

Retail Investing

retail-investing
Retail investing is the act of non-professional investors buying and selling securities for their own purposes. Retail investing has become popular with the rise of zero commissions digital platforms enabling anyone with small portfolio to trade.

Accredited Investor

accredited-investor
Accredited investors are individuals or entities deemed sophisticated enough to purchase securities that are not bound by the laws that protect normal investors. These may encompass venture capital, angel investments, private equity funds, hedge funds, real estate investment funds, and specialty investment funds such as those related to cryptocurrency. Accredited investors, therefore, are individuals or entities permitted to invest in securities that are complex, opaque, loosely regulated, or otherwise unregistered with a financial authority.

Startup Valuation

startup-valuation
Startup valuation describes a suite of methods used to value companies with little or no revenue. Therefore, startup valuation is the process of determining what a startup is worth. This value clarifies the company’s capacity to meet customer and investor expectations, achieve stated milestones, and use the new capital to grow.

Profit vs. Cash Flow

profit-vs-cash-flow
Profit is the total income that a company generates from its operations. This includes money from sales, investments, and other income sources. In contrast, cash flow is the money that flows in and out of a company. This distinction is critical to understand as a profitable company might be short of cash and have liquidity crises.

Double-Entry

double-entry-accounting
Double-entry accounting is the foundation of modern financial accounting. It’s based on the accounting equation, where assets equal liabilities plus equity. That is the fundamental unit to build financial statements (balance sheet, income statement, and cash flow statement). The basic concept of double-entry is that a single transaction, to be recorded, will hit two accounts.

Balance Sheet

balance-sheet
The purpose of the balance sheet is to report how the resources to run the operations of the business were acquired. The Balance Sheet helps to assess the financial risk of a business and the simplest way to describe it is given by the accounting equation (assets = liability + equity).

Income Statement

income-statement
The income statement, together with the balance sheet and the cash flow statement is among the key financial statements to understand how companies perform at fundamental level. The income statement shows the revenues and costs for a period and whether the company runs at profit or loss (also called P&L statement).

Cash Flow Statement

cash-flow-statement
The cash flow statement is the third main financial statement, together with income statement and the balance sheet. It helps to assess the liquidity of an organization by showing the cash balances coming from operations, investing and financing. The cash flow statement can be prepared with two separate methods: direct or indirect.

Capital Structure

capital-structure
The capital structure shows how an organization financed its operations. Following the balance sheet structure, usually, assets of an organization can be built either by using equity or liability. Equity usually comprises endowment from shareholders and profit reserves. Where instead, liabilities can comprise either current (short-term debt) or non-current (long-term obligations).

Capital Expenditure

capital-expenditure
Capital expenditure or capital expense represents the money spent toward things that can be classified as fixed asset, with a longer term value. As such they will be recorded under non-current assets, on the balance sheet, and they will be amortized over the years. The reduced value on the balance sheet is expensed through the profit and loss.

Financial Statements

financial-statements
Financial statements help companies assess several aspects of the business, from profitability (income statement) to how assets are sourced (balance sheet), and cash inflows and outflows (cash flow statement). Financial statements are also mandatory to companies for tax purposes. They are also used by managers to assess the performance of the business.

Financial Modeling

financial-modeling
Financial modeling involves the analysis of accounting, finance, and business data to predict future financial performance. Financial modeling is often used in valuation, which consists of estimating the value in dollar terms of a company based on several parameters. Some of the most common financial models comprise discounted cash flows, the M&A model, and the CCA model.

Business Valuation

valuation
Business valuations involve a formal analysis of the key operational aspects of a business. A business valuation is an analysis used to determine the economic value of a business or company unit. It’s important to note that valuations are one part science and one part art. Analysts use professional judgment to consider the financial performance of a business with respect to local, national, or global economic conditions. They will also consider the total value of assets and liabilities, in addition to patented or proprietary technology.

Financial Ratio

financial-ratio-formulas

WACC

weighted-average-cost-of-capital
The Weighted Average Cost of Capital can also be defined as the cost of capital. That’s a rate – net of the weight of the equity and debt the company holds – that assesses how much it cost to that firm to get capital in the form of equity, debt or both. 

Financial Option

financial-options
A financial option is a contract, defined as a derivative drawing its value on a set of underlying variables (perhaps the volatility of the stock underlying the option). It comprises two parties (option writer and option buyer). This contract offers the right of the option holder to purchase the underlying asset at an agreed price.

Profitability Framework

profitability
A profitability framework helps you assess the profitability of any company within a few minutes. It starts by looking at two simple variables (revenues and costs) and it drills down from there. This helps us identify in which part of the organization there is a profitability issue and strategize from there.

Triple Bottom Line

triple-bottom-line
The Triple Bottom Line (TBL) is a theory that seeks to gauge the level of corporate social responsibility in business. Instead of a single bottom line associated with profit, the TBL theory argues that there should be two more: people, and the planet. By balancing people, planet, and profit, it’s possible to build a more sustainable business model and a circular firm.

Behavioral Finance

behavioral-finance
Behavioral finance or economics focuses on understanding how individuals make decisions and how those decisions are affected by psychological factors, such as biases, and how those can affect the collective. Behavioral finance is an expansion of classic finance and economics that assumed that people always rational choices based on optimizing their outcome, void of context.

Connected Video Lectures

Read Next: Biases, Bounded Rationality, Mandela Effect, Dunning-Kruger

Read Next: Heuristics, Biases.

Main Free Guides:

More Resources

About The Author

Gennaro is the creator of FourWeekMBA, which reached about four million business people, comprising C-level executives, investors, analysts, product managers, and aspiring digital entrepreneurs in 2022 alone | He is also Director of Sales for a high-tech scaleup in the AI Industry | In 2012, Gennaro earned an International MBA with emphasis on Corporate Finance and Business Strategy.
Scroll to Top
FourWeekMBA