Ever wondered how many people might arrive at a hospital’s emergency room in the next hour? Or how likely it is that exactly two calls will hit a customer support line in the next minute?
Poisson Distribution is the answer.
This is the Second major discrete probability distribution students study after the binomial. But unlike the binomial which deals with fixed trials, the Poisson is all about events happening over time, space, or volume especially when they occur rarely but are important to monitor.
Concept
Rare Events Over Time/Space
Poisson Distribution is used to model the number of times an event occurs within a specific interval whether that’s time (per hour), space (per square meter), volume (per litre), or distance (per kilometre).
Think:
- Number of emails received in an hour
- Number of accidents at a junction in a day
- Calls received at a call center per minute
- Defects in a manufacturing line per batch
Assumptions & Key Characteristics
- Events occur one at a time.
- Events are independent (occurrence of one doesn’t affect the other).
- Average rate (λ or lambda) is constant over time/space.
- Two events can’t occur at the exact same instant.
Poisson vs. Binomial
They may seem similar, but they’re not the same. Use the comparison table below:
| Aspect | Binomial | Poisson |
|---|---|---|
| Scenario | Fixed trials | Random time/space interval |
| Parameter | n (trials), p (success prob.) | λ (average rate) |
| Outcome Type | Success/failure | Count of events |
Poisson Formula
The formula for probability of exactly k events in an interval is:
P(X = k) = (e^−λ × λ^k) / k!
Where:
- e is the base of natural log (≈ 2.718)
- λ is the average rate of occurrence
- k is the actual number of events
Mean and Variance
Interestingly, in the Poisson distribution:
- Mean (μ) = λ
- Variance (σ²) = λ
- Standard Deviation = √λ
Example
Let’s say a call center receives on average 3 calls every 2 minutes. What is the probability that it receives exactly 2 calls in a 2-minute span?
Given: λ = 3, k = 2
P(X = 2) = (e^−3 × 3^2) / 2!
≈ (0.0498 × 9) / 2
≈ 0.224
Interpretation: There's approximately a 22.4% chance of receiving exactly 2 calls in the next 2 minutes.
Real-World Application
- Traffic: Number of accidents at a signal per week
- Healthcare: Patient arrivals at an ER per hour
- IT/Service: Server failures or system crashes
- Retail: Customers entering a store per hour
- Manufacturing: Defects in products per batch
Limitations
- Assumes constant rate (λ) real-world rates may vary.
- Only works when the event occurs randomly and independently.
- Not suitable for scenarios with limited/fixed number of trials (use Binomial there).
Summary
- Poisson distribution models rare, random events over time/space.
- Only one parameter: λ (mean = variance = λ).
- Formula uses exponential function and factorials.
- Best used for queues, arrivals, accidents, and rare occurrences.
- Key in analytics, operations, logistics, and customer service planning.
Final Thought: The next time you’re stuck in traffic or waiting on hold, ask yourself could this be modeled with a Poisson distribution? You’ll be surprised how often the answer is yes.