16.1 Probability Distribution Function (PDF) of Poisson Distribution

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. It is particularly useful when these events satisfy the following conditions:

• Independence: The occurrence of one event does not affect the probability of another event occurring.
• Constant Average Rate: The events occur at a constant average rate over the specified interval.
• Infrequent or Rare: The events are relatively infrequent or rare within the interval.

Practical Scenario

Consider a bakery that receives an average of 4 customers per hour. The Poisson distribution can be used to calculate the probability of receiving a specific number of customers within that one-hour period. For instance, we could find the probability of receiving exactly 5 customers.

Poisson PDF Formula

The probability mass function (PMF) for the Poisson distribution is given by:

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

Where:

• $P(X = x)$: The probability of observing exactly $x$ events.
• $\lambda$ (lambda): The average number of events (rate of occurrence) within the fixed interval.
• $e$: Euler's number, the base of the natural logarithm (approximately 2.71828).
• $x!$: The factorial of $x$ (e.g., $3! = 3 \times 2 \times 1 = 6$).

Valid for: $x = 0, 1, 2, 3, \dots$ (non-negative integers) and $\lambda > 0$.

Explanation of Parameters

• $\lambda$ (lambda): This parameter represents the average rate at which events occur in the specified interval of time or space. For example, if a call center receives an average of 10 calls per minute, then $\lambda = 10$.
• $x$: This is the number of actual events observed or of interest within that interval. For example, if we want to know the probability of receiving exactly 3 calls in a minute, then $x = 3$.
• $e$: Euler's constant, a fundamental mathematical constant approximately equal to 2.71828. It arises naturally in many areas of mathematics, including calculus and probability.
• $x!$ (Factorial of $x$): The product of all positive integers up to $x$. For example, $5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$. By definition, $0! = 1$.

Application Areas

The Poisson distribution is widely applied in various fields:

• Call Centers: Modeling the number of calls received per minute or hour.
• Web Traffic: Analyzing the number of website visits or data packets per second.
• Biology: Calculating the occurrences of mutations, cell divisions, or disease outbreaks in a given period.
• Retail: Predicting customer arrivals at a store or checkout counter.
• Manufacturing: Tracking the number of defects on a production line per unit area or time.
• Insurance: Estimating the number of claims made by policyholders in a year.

Interview Questions

1. What are the key conditions under which the Poisson distribution is applied?
   • Events occur independently.
   • Events occur at a constant average rate.
   • Events are rare or infrequent.
2. Explain the role of $\lambda$ (lambda) in the Poisson distribution.
   • $\lambda$ represents the average rate or expected number of events in a fixed interval of time or space. It is the single parameter that defines the entire distribution.
3. Derive or explain the Poisson probability formula.
   • The Poisson distribution can be derived as a limiting case of the Binomial distribution when the number of trials ($n$) is very large and the probability of success ($p$) in each trial is very small, such that $\lambda = np$ remains constant. The formula arises from this limiting process and represents the probability of observing $x$ events given an average rate $\lambda$.
4. Why is the Poisson distribution suitable for rare or infrequent events?
   • The conditions of independence and constant average rate, combined with the nature of the formula which gives a higher probability for smaller counts of events when $\lambda$ is small, makes it appropriate for scenarios where events don't happen all the time but occur randomly over time or space.
5. How do you compute the probability of observing exactly $x$ events using Poisson?
   • By plugging the values of $x$ (the number of events), $\lambda$ (the average rate), and $e$ (Euler's number) into the Poisson PMF formula: $P(X = x) = (\lambda^x * e^{-\lambda}) / x!$.
6. Provide a real-life example where Poisson distribution is applied.
   • A common example is modeling the number of customers arriving at a bank teller window per hour, assuming the arrivals are independent and occur at a relatively constant average rate.
7. What is the significance of Euler’s number ($e$) in the Poisson formula?
   • Euler's number ($e$) is intrinsic to the exponential function $e^{-\lambda}$, which is crucial for normalizing the distribution and ensuring that the probabilities sum to 1. It reflects the continuous nature of the underlying process that gives rise to the discrete Poisson counts.
8. How do you determine if Poisson is the right model for your data?
   • Check if the data counts discrete events in a fixed interval. Assess if the events are independent and occur at a roughly constant rate. Compare the mean and variance of your data; for a Poisson distribution, the mean should be approximately equal to the variance.
9. How does the Poisson distribution differ from the Binomial distribution?
   • Binomial: Deals with a fixed number of trials ($n$), each with two outcomes (success/failure), and a constant probability of success ($p$) for each trial. It's about the number of successes in $n$ trials.
   • Poisson: Deals with the number of events occurring in a fixed interval, where the number of trials is not fixed or is infinite, and the probability of an event in any given sub-interval is very small. It's about the count of events over a continuous interval.
10. In which industries or fields is the Poisson distribution most useful?
    • Telecommunications, customer service, quality control, queuing theory, finance, epidemiology, traffic management, and anywhere that involves counting rare or infrequent events occurring randomly over time or space.