Enter the average rate of success (λ) and a Poisson random variable (x) to calculate all key probabilities for the Poisson distribution. You get P(X=x), P(X
P(X = x)
--
P(X < x)
--
P(X ≤ x)
--
P(X > x)
--
P(X ≥ x)
--
Mean (μ)
--
Variance (σ²)
--
Standard Deviation (σ)
--
The Poisson distribution is a discrete probability distribution that models the number of times an event occurs within a fixed interval of time or space, given a known average rate (λ). It assumes events occur independently and at a constant average rate. It's commonly used in scenarios like call center arrivals, radioactive decay counts, or website traffic.
Lambda (λ) is the expected number of events in the given interval — for example, if a call center receives an average of 4 calls per hour, then λ = 4. Lambda must be a positive number. It also equals the mean and variance of the Poisson distribution.
The Poisson random variable x is the specific count of events you want to evaluate the probability for. It must be a non-negative integer (0, 1, 2, …). For instance, you might ask: what is the probability of exactly 3 calls arriving in an hour when the average is 4?
The probability of exactly x events occurring is P(X = x) = (e^−λ × λˣ) / x!, where e ≈ 2.71828 is Euler's number, λ is the average rate, and x! is the factorial of x. Cumulative probabilities like P(X ≤ x) are computed by summing P(X = k) for all k from 0 up to x.
A cumulative Poisson probability is the probability that the random variable X takes a value within a specified range. P(X ≤ x) sums all probabilities from 0 to x (left-tail/CDF), while P(X ≥ x) covers x and above (right-tail). These are useful when you want to know the likelihood of 'at most' or 'at least' a given number of events.
For a Poisson distribution, the standard deviation is σ = √λ. Since the variance equals λ, the standard deviation is simply the square root of the average rate. For example, if λ = 9, then σ = 3.
The Poisson distribution applies whenever you're counting rare, independent events over a fixed interval. Classic examples include: the number of customer arrivals per hour at a store, the number of defects in a manufacturing batch, the number of emails received per day, and the number of earthquakes in a region per year.
A Poisson experiment is a statistical experiment with the following properties: the outcomes can be classified as successes or failures; the average number of successes (λ) is known; events occur independently of one another; and the probability of success in any tiny sub-interval is proportional to the length of that sub-interval.