Poisson Distribution Calculator

What the Poisson Distribution Calculator Does

The Poisson Distribution Calculator helps you compute event-count probabilities when outcomes are modeled as counts over a fixed interval. Instead of looking up values in tables or manually summing many terms, you can calculate the exact probability of observing k events, cumulative probabilities up to a threshold, right-tail risk (at least k events), and probability within a range of counts. The tool also includes inverse CDF (quantile) so you can find the cutoff count that corresponds to a given percentile.

The Poisson distribution is widely used for counting events like arrivals, calls, defects, transactions, website hits, accidents, or rare failures—especially when the average rate is known and events are assumed to occur independently. If your problem sounds like “How likely is it to see 0, 1, 2, … events in this time window?” then the Poisson model is often the first place to start.

Poisson Distribution Basics: λ, Mean, and Variance

A Poisson distribution is parameterized by λ (lambda), the expected number of events in your chosen interval. If λ=3, you expect about 3 events per interval on average. A special property of the Poisson distribution is that: mean = λ and variance = λ. That means the standard deviation is √λ.

This mean-variance equality is useful for quick sanity checks. If your observed count data has variance much larger than its mean (a common real-world pattern), the data may show overdispersion and a plain Poisson model may underestimate tail risk. In that case, models like negative binomial or a Poisson mixture may be more appropriate. But if mean and variance are reasonably close, Poisson can be a strong fit.

Rate × Time: How to Build λ from Real Inputs

In practice, you often know an event rate per unit time rather than λ directly. For example, “2 calls per minute” or “0.3 failures per month.” The Poisson model commonly uses:

This calculator supports both input styles. If you choose Compute λ = rate × time, the tool multiplies your rate by your time window and uses that λ in all probability calculations. This makes it easy to model “expected events in 10 minutes” when the base rate is “events per minute.”

PMF: Probability of Exactly k Events

The most direct Poisson question is: “What is the probability of exactly k events in this interval?” That is the PMF:

Here, k is a nonnegative integer (0, 1, 2, …). The factorial k! grows quickly, which is why manual calculation becomes tedious. This calculator handles those terms safely and returns the PMF value in decimal, percent, or scientific notation.

Interpreting PMF is simple: it is the probability of landing on that exact count. If you want the probability of at most or at least a value, you need cumulative probability (CDF) or tail probability.

CDF and Tail Probabilities: “At Most” and “At Least”

The CDF answers “at most” questions:

In words, the CDF is the sum of PMF terms up to k. The calculator computes this sum and also shows right-tail values. Right tails (“at least k”) are often more important in risk and capacity planning:

The calculator’s CDF & Tails mode lets you choose P(X ≤ k), P(X < k), P(X ≥ k), or P(X > k), so you can match the exact wording of your question without off-by-one mistakes.

Between Probabilities: Ranges of Counts

Many real decisions depend on ranges rather than exact values. Examples include: “between 2 and 6 incidents,” “within the allowed defect limit,” or “a typical day has 10–20 customers.” For a Poisson random variable, the inclusive between probability is:

This is exactly what the calculator’s Between tab does. You can also choose exclusive or half-open intervals to match how your counts are defined. For example, “more than a but at most b” corresponds to (a, b].

Inverse CDF: Finding Poisson Cutoffs and Percentiles

Sometimes you know the probability target and want the threshold count. This is the inverse CDF (quantile): “Find the smallest k such that P(X ≤ k) ≥ p.” Poisson quantiles are always integers because counts are discrete.

Quantiles are useful for planning and decision rules:

• Capacity: “How many arrivals should we be prepared for 95% of the time?”
• Alerts: “If counts exceed this threshold, it’s unusual (upper 1% tail).”
• Service levels: “Pick a limit so that overflow happens only 5% of days.”

This calculator supports both left-tail percentiles and a right-tail style cutoff (tail risk framing). It also reports a check value for P(X≤k) and P(X≥k) so you can validate the cutoff immediately.

Poisson Assumptions and When They Matter

The Poisson distribution is not a universal count model; it works best when a few assumptions are approximately true:

• Independent events: one event does not make another more or less likely in the same interval.
• Constant rate: the average rate doesn’t change substantially within the interval.
• Rare in tiny sub-intervals: probability of multiple events in a very small slice of time is negligible.

Violations can show up in your data as overdispersion, bursts/clustering, or strong time-of-day seasonality. If you see counts that “come in waves,” a Poisson model may underestimate the probability of extreme days. In those cases, consider splitting the interval into smaller segments with different rates, or using a model that handles variable rates.

Poisson vs Binomial vs Normal

The Poisson distribution is closely related to other classic distributions:

• Binomial limit: when n is large and p is small with np=λ fixed, Binomial(n,p) ≈ Poisson(λ).
• Normal approximation: when λ is moderately large, Poisson(λ) is often approximated by Normal(μ=λ, σ=√λ).

The normal approximation can be convenient for quick mental estimates and for building intuition, but because Poisson is discrete, normal approximations can misestimate probabilities for small λ or near the boundary at 0. For exact answers—especially in tail probabilities—use the Poisson formulas.

How to Use This Calculator in Practice

1) Decide the interval and compute λ

The “interval” can be time (per hour/day/month) or an exposure unit (per product batch, per kilometer, per customer session). If your rate is r per unit and the exposure is t units, use λ=r×t. If you already have an expected count per interval, enter λ directly.

2) Pick the question type

Choose the tab that matches your question:

• PMF: probability of exactly k events.
• CDF & Tails: “at most” or “at least” threshold probabilities.
• Between: probability that counts fall within a range.
• Inverse CDF: find the count cutoff for a probability target.
• Table: view a full probability table from 0 to max k (plus expected counts if you provide n).

3) Interpret the output as a decision tool

Poisson outputs are best interpreted as long-run frequencies under the model assumptions. If P(X ≥ 10)=0.02, then under a stable Poisson process, you’d expect 10 or more events about 2% of intervals. This is commonly used to set alert thresholds (“unusual if above this count”) or to plan resources (“size capacity so overflow is rare”).