What the Poisson Distribution Models
The Poisson distribution models the number of events occurring in a fixed interval of time, space, distance, area, or volume. It is useful when events happen independently and at a constant average rate.
Here, λ is the expected average number of events in the interval, and k is the event count.
What Lambda Means
Lambda is not a percentage. It is an expected count. If a call center receives an average of 12 calls per hour, then λ = 12 for a one-hour interval. For a half-hour interval under the same rate, λ = 6.
Probability Types
| Mode | Meaning | Formula idea |
|---|---|---|
| Exact | Exactly k events. | P(X = k) |
| Less than | Fewer than k events. | P(X < k) = P(X ≤ k − 1) |
| At most | k or fewer events. | P(X ≤ k) |
| At least | k or more events. | P(X ≥ k) = 1 − P(X ≤ k − 1) |
| More than | More than k events. | P(X > k) = 1 − P(X ≤ k) |
| Between | Between k and b events, inclusive. | P(k ≤ X ≤ b) |
Why This Calculator Avoids Direct Factorials
The textbook formula is easy to read, but direct calculation of λk and k! can become unstable for large k. This calculator uses logarithms and log-sum-exp for range probabilities.
This is more stable than directly forming very large powers and factorials first.
Example: Email Arrivals
Suppose you receive an average of 4 emails per hour and want the probability of exactly 2 emails in the next hour.
λ = 4 k = 2 P(X = 2) = (4² × e⁻⁴) / 2! P(X = 2) ≈ 0.1465
Mean, Variance, and Standard Deviation
The Poisson distribution has a simple relationship between its average and spread.
Variance = λ
Standard Deviation = √λ
When a Poisson Model Makes Sense
- Events are counted in a fixed interval.
- Events occur independently.
- The average rate is approximately constant.
- Two events are unlikely to occur at exactly the same instant in very small intervals.
Common Uses
- Emails received per hour.
- Phone calls received by a call center.
- Cars arriving at a toll booth.
- Machine failures over a time interval.
- Defects per manufactured unit.
- Website visits per minute.
- Accidents at an intersection over a fixed period.
Common Mistakes
- Using a negative value for k. Poisson counts cannot be negative.
- Using a non-integer value for k. Poisson event counts are whole numbers.
- Mixing time intervals, such as λ per hour with k per minute.
- Using Poisson when events are not independent.
- Using Poisson when the event rate changes strongly over the interval.
Frequently Asked Questions
What is the Poisson distribution used for?
It is used to model event counts over a fixed interval when events occur independently at a constant average rate.
Can k be decimal?
No. k must be a non-negative integer because it represents a count of events.
Can lambda be decimal?
Yes. Lambda can be decimal. For example, λ = 2.5 means the expected average count is 2.5 events per interval.
What is the mean and variance of a Poisson distribution?
For a Poisson distribution, the mean is λ and the variance is also λ.
What is the difference between Poisson and binomial distributions?
The binomial distribution counts successes in a fixed number of trials. The Poisson distribution counts events in a fixed interval when events happen at an average rate.