The Poisson Probability Distribution

Statistics • Discrete Random Variables and Their Probability Distributions

Written by STEM Calculators Team Published December 13, 2025 Updated February 24, 2026

The Poisson probability distribution

Use this tool when a discrete random variable X counts the number of occurrences in a fixed interval and the occurrences are random and independent. Enter the mean rate λ for the same interval you are counting.

Quick checklist (Poisson conditions)

X is a discrete random variable (0, 1, 2, …).
Occurrences are random (no predictable pattern within the interval).
Occurrences are independent (one does not affect another).

Parameters

λ = mean number of occurrences in the interval.
x = actual number of occurrences (the value taken by X).

Context (optional)

This is only used to label the result; it does not affect calculations.

Mean rate (λ) (occurrences per interval)

λ must match the same interval you count for X. If not, use the conversion below.

Probability query

“At most” includes k; “less than” excludes k.

k (nonnegative integer)

Use whole numbers only: 0, 1, 2, …

Convert λ from a different interval

Example: if the mean is given “per 10 items” but you count “per 40 items”, scale λ by 40/10.

Animate chart

Smoothly grows the bars after each calculation.

Chart/table maximum x (optional)

If blank, the tool chooses a reasonable range around λ and your selected bounds.

Ready

Enter values and click “Calculate”.

Visualization

Bars show the probability of each count x. The shaded region matches the probability query you selected.

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The Poisson probability distribution

The Poisson distribution models a count of occurrences in a fixed interval (time, space, area, volume, items), when occurrences are random and independent and the mean rate is constant over the interval.

When it is appropriate

X is a discrete random variable: 0, 1, 2, …
Occurrences are random (unpredictable within the interval).
Occurrences are independent (one occurrence does not affect another).

Typical examples include counts of calls received, defects found, customers arriving, accidents occurring, or items returned.

Parameter and notation

Let λ be the mean number of occurrences in the interval, and let x be the actual number of occurrences (the value taken by X).

Poisson probability formula

The probability of exactly x occurrences is given by the Poisson PMF:

\[ \begin{aligned} P(X = x) &= \frac{\lambda^{x}\cdot e^{-\lambda}}{x!}, \quad x=0,1,2,\dots \end{aligned} \]

Common probability queries

Many questions ask for cumulative probabilities. These are built by summing Poisson PMF terms.

\[ \begin{aligned} P(X \le k) &= \sum_{x=0}^{k}\frac{\lambda^{x}\cdot e^{-\lambda}}{x!} \\ P(X < k) &= \sum_{x=0}^{k-1}\frac{\lambda^{x}\cdot e^{-\lambda}}{x!} \\ P(X \ge k) &= 1 - P(X \le k-1) \\ P(X > k) &= 1 - P(X \le k) \end{aligned} \]

Important interval note

The counting interval for X must match the interval used to define λ. If the mean is given for a different interval, convert it first:

\[ \begin{aligned} \lambda_{\text{target}} &= \lambda_{\text{given}} \cdot \frac{\text{target interval}}{\text{given interval}} \end{aligned} \]

Example idea: if a mean is “per 10 items” but you count “per 40 items”, multiply λ by 40/10.

Short worked example (pattern)

Step 1. Identify λ and the event.

Suppose the mean is λ = 3 per interval and we want “at most 1 occurrence”.

Step 2. Use the cumulative definition.

\[ \begin{aligned} P(X \le 1) &= P(X=0) + P(X=1) \\ &= \frac{\lambda^{0}\cdot e^{-\lambda}}{0!} + \frac{\lambda^{1}\cdot e^{-\lambda}}{1!} \end{aligned} \]

The calculator automates the factorial, exponential, and the needed sums, and it visualizes the PMF as a bar chart so you can see which counts contribute to the selected event.

Frequently Asked Questions

What does lambda mean in the Poisson distribution?

Lambda is the mean (expected) number of occurrences in the interval being counted. It must match the same interval as the random variable X (for example, per hour if X counts events per hour).

How do I compute P(X <= k) with this Poisson calculator?

Choose the "At most k" option and enter k. The calculator sums Poisson PMF terms from x = 0 up to x = k to produce the cumulative probability.

What is the difference between "at most k" and "less than k"?

At most k means P(X <= k), which includes k. Less than k means P(X < k), which excludes k and sums only up to k - 1.

How do I convert lambda from one interval to another?

Use the conversion section by entering the given mean lambda_given, the base size, and the target size. The calculator scales the rate as lambda = lambda_given * (target/base) so the mean matches the counting interval.

When is the Poisson distribution appropriate for event counts?

It is appropriate when X is a discrete count (0, 1, 2, ...), occurrences are random within the interval, and occurrences are independent. It is commonly used for counts like arrivals, defects, or incidents over a fixed period or region.

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The Poisson probability distribution

Quick checklist (Poisson conditions)

Parameters

Visualization

Calculation steps

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Frequently Asked Questions

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