What Is a Poisson Random Variable?
The Poisson random variable models the number of times an event occurs in a fixed interval of time, space, or volume, given that these events occur randomly and independently, and at a constant average rate.
Mathematical Definition
If X is a Poisson random variable with average rate λ, then:
P(X = k) = (e-λ * λk) / k!
λ: the average number of eventsk: number of occurrences (0, 1, 2, …)e: Euler’s number (~2.718)
Key Properties
- Type: Discrete
- Parameter: λ > 0
- Mean = λ
- Variance = λ
- Skewed right (more extreme values on the right)
When to Use It
Use the Poisson distribution when:
- You’re counting events (not measuring)
- Events are rare, random, and independent
- You’re working with a fixed interval of time or space
- Expected value (mean):
E(X) = λ - Variance:
Var(X) = λ
Expectation and Variance
One of the unique properties of the Poisson distribution is that its mean (expected value) and variance are both equal to λ, the average rate of occurrence.
This means if you’re observing an event that occurs on average 4 times in a fixed interval (i.e., λ = 4), then:
E(X) = 4, Var(X) = 4
Intuitively, this tells us that not only do we expect around 4 events per interval, but the spread (dispersion) of the data is also quantified by that same number.
As λ increases, the Poisson distribution becomes more symmetric and starts resembling a normal distribution. This is due to the Central Limit Theorem, and it’s particularly useful when approximating probabilities for large λ values.
Detailed Examples
1. Web Server Load
Suppose a server receives 2 requests per second. Let X be the number of requests per second. What is P(X < 5)?
That means we sum: P(X = 0) to P(X = 4). We add these because each value (0,1,…,4) is mutually exclusive — the server can’t receive 2 and 3 hits at once.
- P(0) = 0.1353
- P(1) = 0.2707
- P(2) = 0.2707
- P(3) = 0.1804
- P(4) = 0.0902
Total ≈ 0.9473. So there’s a 94.7% chance of getting fewer than 5 hits per second.
2. ER Visits
A hospital gets 3 patients per hour on average. What’s the probability of exactly 2 patients arriving?
λ = 3, k = 2:
P(X=2) = (e^-3 * 3²) / 2! = (0.0498 * 9) / 2 ≈ 0.2240
3. Call Center Silence
A call center receives 10 calls per minute. What’s the probability that no calls are received in a minute?
λ = 10, k = 0:
P(X=0) = (e^-10 * 10^0) / 0! = e^-10 ≈ 0.000045
4. Rainfall in the Desert
It rains 1 day/week on average in a desert town. What’s the probability of exactly 2 rainy days this week?
λ = 1, k = 2:
P(X = 2) = (e^-1 * 1^2) / 2! = 0.3679 / 2 ≈ 0.1839
Final Thoughts
The Poisson distribution is powerful in modeling rare, countable events. Whether it’s network traffic, patient visits, or environmental events, mastering Poisson equips you to think statistically and make better decisions.