A yes-or-no outcome, a success count, and an event count require different discrete models.
Whether one customer purchases is a success-or-failure outcome. The number of purchasers among 100 customers is a success count. The number of support requests in one hour is an event count.
All three involve counts, but the data-generating structures differ.
Key question
Is the variable one binary outcome, a fixed number of repeated successes, or a count within time or space?
How are the three distributions connected?
Bernoulli
One success-or-failure trial
X ∈ {0,1}
Binomial
Number of successes in n independent trials
X ∈ {0,…,n}
Poisson
Number of events in a fixed interval
X ∈ {0,1,2,…}
Repeating independent Bernoulli trials n times with the same success probability produces a binomial count. A Poisson variable counts events within an interval under a stable average rate.
When do Bernoulli and binomial models apply?
| Distribution | Mean | Variance |
|---|---|---|
| Bernoulli(p) | p | p(1−p) |
| Binomial(n,p) | np | np(1−p) |
Binomial example
The number of purchasers among 50 people exposed to an advertisement
What situations does the Poisson model describe?
A Poisson model describes how many events occur in a fixed interval of time or space. λ is the expected number of events in that interval.
| Example | Interval | Event |
|---|---|---|
| Call center | one hour | incoming requests |
| Server | one day | errors |
| Road | one kilometre | accidents |
A defining feature
The Poisson mean and variance are both λ. If the observed variance is much larger, overdispersion may be present.
How should a distribution be chosen?
| Question | Distribution |
|---|---|
| Did one trial succeed? | Bernoulli |
| How many successes in 100 trials? | Binomial |
| How many events occurred in one hour? | Poisson |
Poisson approximation to the binomial
When n is large and p is small, a binomial distribution can be approximated by Poisson with λ=np.
Change the shapes of binomial and Poisson distributions
Adjust n, p, and λ to see how the center and spread change.
Mean
3.00
Variance
2.10
Key lesson
A distribution’s shape reflects the rule that generates the data, not merely a visual pattern.