Chapter 22
THE POISSON DISTRIBUTION
Clear and complete explanation of the general aspects of the Poisson distribution , useful for articles, teaching, or statistical analysis:
Generalities
The Poisson distribution is a discrete probability distribution that describes the frequency with which an event occurs in a fixed interval of time or space , provided that :
Events occur independently of each other.
The rate of occurrence (λ, "lambda") is constant .
There is no more than one event occurring at the exact same time.
Common uses: Used when you want to model the number of times an event occurs in a given period or space. Some examples:
Number of calls received by a central office in an hour.
Number of typographical errors on a printed page.
Arrival of buses at a station at a certain time.
Number of radioactive particles detected in one second.
Distribution formula
The probability function of the Poisson distribution is:
P(X=k)=e−λλkk!P(X = k) = \frac{e^{-λ} λ^k}{k!}P(X=k)=k!e−λλk
Where:
P(X=k)P(X = k)P(X=k) is the probability that exactly kkk events occur.
λλλ is the average number of occurrences (expected).
eee is Euler's constant (≈ 2.718).
kkk is the number of events (0, 1, 2, …).
!k!k! is the factorial of kkk.
Important features
Mean : E(X)=λE(X) = λE(X)=λ
Variance : Var(X)=λVar(X) = λVar(X)=λ
Asymmetric distribution (skewed to the right) when λλλ is small.
It tends to a normal distribution when λλλ is large (central limit theorem).
Key points to remember: Ideal for rare events, there is no upper limit to the number of events.
Requires events to be independent and random .
Practical example: Call prediction in a customer service center using AI and Poisson distribution
Situation: A telecommunications company wants to predict how many calls its call center will receive in 10-minute intervals to optimize agent allocation using an artificial intelligence system.
It has been determined that, on average, 3 calls are received every 10 minutes .
Application of the Poisson distribution:
The AI uses the Poisson distribution with:
λ=3λ = 3λ=3 (average call rate per 10 minutes).
We want to know, for example, what is the probability that exactly 5 calls arrive in a 10-minute interval?
P(X=5)=e−3 ⋅ 355!=e−3 ⋅ 243120≈0.
So there is a 10.08% chance of receiving exactly 5 calls in 10 minutes.
How does intelligence come in?
ChatGPT said:
Excellent question, José Saul! Here's how artificial intelligence fits into this context, connecting with the example of the customer service center:
AI doesn't replace the Poisson distribution, the power distribution . Here's how they work:
Machine Learning
AI analyzes large volumes of historical data : times, dates, number of calls, duration, reasons, etc. With this information, it automatically estimates the optimal λλλ value for different times of day.
So instead of a single average value, AI can learn that:
At 8:00 am there is a λ = 2 (low load).
At noon, λ = 6 (peak hour).
At 4:00 pm, λ = 4.
Real-time prediction: AI uses the Poisson distribution to predict, for example:
The probability that 0, 1, 2, ..., 10 calls will come in in the next 10 minutes.
This information is used to assign the appropriate number of agents and avoid overload or inactivity.
Intelligent Adaptation (Dynamic AI): The system can:
detect patterns by day of the week or by season, automatically adjust the model if a holiday changes behavior.
Combine the Poisson distribution with neural network algorithms that detect non-obvious patterns.
Integration into management systems: Intelligent platforms (such as chatbots, virtual assistants, or CRM systems ) use these predictions to prioritize urgent calls and trigger automated messages when high demand is expected.
Suggest off-peak times to users.
Conclusion
AI does not replace the Poisson distribution , but rather:
Learn the best value of λλλ,
It is applied in a personalized way by hour or event and
use that data to make automated decisions in real time .
AI AND POISSON DISTRIBUTION: PREDICTING CALLS IN A CALL CENTER
Statistics and artificial intelligence (AI) are merging increasingly powerfully into everyday life. A practical and useful example is the use of the Poisson distribution in combination with AI systems , especially in the service sector.
A common challenge: How many calls will arrive?
Let's imagine a telecommunications company that wants to predict how many calls will come into its customer service center in 10-minute periods. This data is vital to ensuring that there are neither too many idle agents nor an overload of unanswered calls.
Historically, it's known that an average of three calls arrive every 10 minutes. These types of situations—events that occur randomly but with a certain regularity—are ideal for applying the Poisson distribution .
The Poisson distribution is a mathematical model that calculates the probability of kkk events occurring in a given time or space interval, with an average rate λλλ. For example, the probability of exactly 5 calls arriving in a 10-minute period can be calculated if λ=3λ = 3λ=3:
P(X=5)=e−3 ⋅ 355!≈0.1008P(X = 5) = \frac{e^{-3} \cdot 3^5}{5!} ≈ 0.1008P(X=5)=5!e−3 ⋅ 35≈0.1008
That is, there is a 10.08% chance of receiving exactly
How does artificial intelligence come in?
AI is responsible for taking this model to the next level:
Learn from history : AI analyzes past data to detect patterns and adjust lll by time of day, day of week, or season.
Predicts in real time : Thanks to machine learning, the system can anticipate peaks or drops in calls with high accuracy.
Make smart decisions : Based on these predictions, you can trigger automated messages, redeploy agents, or schedule alerts for your supervisor.
Adapts : If something unusual happens (like an unexpected holiday or ad campaign), the AI adjusts its predictions in real time.
Conclusion: The combination of classic models such as the Poisson distribution with the adaptive capabilities of artificial intelligence results in modern, efficient, and automated solutions. This combination represents an important step toward resource optimization for companies across all sectors.
