Math
Poisson Distribution Calculator
Compute Poisson probabilities for exactly, at most, at least, or a range of events. Returns PMF, CDF, mean, variance, and a distribution preview.
Probability of X events in a fixed interval
Mean number of events per interval. Must be greater than 0.
P(X = 2)
0.224042
22.4042%
Probability mass at exactly k events.
P(X = k)
0.224042
P(X <= k)
0.42319
P(X > k)
0.57681
P(X >= k)
0.800852
Mean
3
Variance
3
Std deviation
1.7321
Mode
2
Distribution preview (k = 0 to 20)
| k | P(X = k) | Shape |
|---|---|---|
| 0 | 0.049787 | |
| 1 | 0.149361 | |
| 2 | 0.224042 | |
| 3 | 0.224042 | |
| 4 | 0.168031 | |
| 5 | 0.100819 | |
| 6 | 0.050409 | |
| 7 | 0.021604 | |
| 8 | 0.008102 | |
| 9 | 0.002701 | |
| 10 | 0.00081 | |
| 11 | 0.000221 | |
| 12 | 5.52e-5 | |
| 13 | 1.27e-5 | |
| 14 | 2.73e-6 | |
| 15 | 5.46e-7 | |
| 16 | 1.02e-7 | |
| 17 | 1.81e-8 | |
| 18 | 3.01e-9 | |
| 19 | 4.76e-10 | |
| 20 | 7.14e-11 |
Frequently Asked Questions about the Poisson Distribution Calculator
What is the Poisson distribution?
The Poisson distribution models the count of independent events that occur at a constant average rate within a fixed interval of time, space, or volume. It is fully described by a single parameter, lambda, which is both the mean and the variance. Examples include calls per hour, emails per day, or radioactive decays per second.
What is the formula for the Poisson PMF?
P(X = k) = (lambda^k x e^(-lambda)) / k!, where lambda is the average rate and k is a non-negative integer. The CDF P(X <= k) sums the PMF from i = 0 to i = k. This calculator uses the log-space form so probabilities stay accurate even when lambda or k are large.
When should I use Poisson instead of binomial?
Use Poisson when events are rare and the number of opportunities is large, but you do not know either number directly, only the average rate. Binomial needs a fixed n and a per-trial probability p. As n grows and p shrinks with n*p constant, binomial converges to Poisson with lambda = n*p.
What are the mean, variance, and mode of a Poisson distribution?
Mean and variance both equal lambda, so the standard deviation is sqrt(lambda). The mode is floor(lambda) when lambda is not an integer. When lambda is a whole number, both lambda - 1 and lambda are tied as the most likely outcomes.
How do I read the at most, at least, and range modes?
At most k computes P(X <= k). At least k computes P(X >= k) = 1 - P(X <= k - 1). Range computes P(k1 <= X <= k2) = P(X <= k2) - P(X <= k1 - 1).