Poisson Probability Calculator

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Poisson Distribution Calculator

This Poisson Distribution Calculator determines probabilities for a given number of events occurring in a fixed interval, along with mean, variance, and standard deviation.

Poisson Distribution Calculator

Enter a Poisson mean and choose a probability query to inspect exact counts, tails, or intervals.

Poisson Mean (lambda):

Probability Query:

Count (k):

Select a query to calculate one focused probability and highlight the matching outcomes.

Related Calculators:

Normal Distribution Binomial Distribution

Calculator Resources

This section provides essential information and guidance to help you make the most of the calculator.

What is the Poisson Distribution?

The Poisson distribution is a discrete probability distribution that models the probability of a given number of events occurring in a fixed interval of time or space. It is particularly useful for situations where events happen independently of each other and at a constant average rate (denoted as $\lambda$, also written as $\mu$). This distribution is often applied to rare events that occur over a continuous period or within a defined area.

Purpose of the Calculator

This Poisson Distribution Calculator is a practical tool for analyzing count probabilities. It computes exact, lower-tail, upper-tail, and interval probabilities, including open and closed interval endpoints. Interactive PMF and CDF charts, role-aware table highlights, and key descriptive statistics help connect the numbers to the distribution's shape.

Probability Query Modes: Calculates exact queries such as $P(X = k)$, inequalities such as $P(X \leq k)$ and $P(X \geq k)$, and interval queries such as $P(a < X \leq b)$.
Cumulative Distribution Function (CDF): Computes $P(X \leq k)$ (the probability of $k$ or fewer events) and $P(X \geq k)$ (the probability of $k$ or more events), showing the lower and upper tails around a selected count.
Descriptive Statistics: Instantly displays the mean ($\mu$ or $\lambda$), variance ($\sigma^2$), and standard deviation ($\sigma$) of the Poisson distribution based on the input average rate.
Visualization: Generates clear and interactive PMF and CDF charts, allowing for a visual interpretation of the probability distribution's shape and cumulative behavior.
Table Output: Presents a comprehensive probability table showing $P(X = k)$, $P(X \leq k)$, and $P(X \geq k)$ for a range of possible $k$ values, providing a detailed overview of the distribution.
Role-aware Highlighting: Highlights PMF outcome cells, CDF endpoints used in subtraction, and direct upper-tail cells so the table shows which values actually enter the calculation.
Flexible Input: Accommodates a positive mean ($\mu$ or $\lambda$) within the stable calculation range and any non-negative integer $k$ value for calculations.

Enter the Poisson mean ($\mu$ or $\lambda$):
This value represents the average rate of events occurring in the fixed interval of time or space. It must be a positive number.
Choose a probability query:
Select an exact, inequality, or interval query. Single-value queries use $k$; interval queries use bounds $a$ and $b$.
Enter the required count value or bounds:
Use a non-negative integer for $k$, $a$, and $b$. The symbols in the query label show whether each endpoint is included.
Click "Compute":
The calculator will process the input and display the calculated probabilities, descriptive statistics, and interactive charts.
View the PMF and CDF charts, probability table, and detailed output:
Examine the graphical representations to understand the distribution's shape and the table for precise probability values across different $k$.
(Optional) Highlight/hide specific features or switch between PMF and CDF charts:
Utilize the interface controls to customize your view of the results, such as focusing on a specific $k$ or toggling between the different chart types.

Probability Mass Function (PMF):

The probability of observing exactly $k$ events in a fixed interval:

$$P(X = k) = \frac{\lambda^k \, e^{-\lambda}}{k!}$$

$\lambda$ is the average rate of events per interval (the Poisson mean).
$k$ is the actual number of events observed (a non-negative integer).
$e$ is Euler's number (approximately 2.71828).
$k!$ is the factorial of $k$.

Cumulative Probability:

$P(X \leq k)$: The probability of observing $k$ or fewer events (the CDF). Calculated as the sum of the PMFs from $i = 0$ to $k$:

$$P(X \leq k) = \sum_{i=0}^{k} \frac{\lambda^i \, e^{-\lambda}}{i!}$$

$P(X \geq k)$: The probability of observing $k$ or more events.

$$P(X \geq k) = 1 - P(X < k) = 1 - P(X \leq k-1)$$

Interval probabilities use differences of CDF endpoints. For example:

$$P(a < X \leq b) = F(b) - F(a)$$

Mean, Variance, and Standard Deviation:

Mean ($\mu$): $\lambda$
Variance ($\sigma^2$): $\lambda$
Standard Deviation ($\sigma$): $\sqrt{\lambda}$

A unique property of the Poisson distribution is that its mean and variance are always equal.

PMF Table

The table provides a detailed breakdown for each possible number of events ($k$):

$P(X = k)$: The probability of observing exactly $k$ events.
$P(X \leq k)$: The cumulative probability of observing up to and including $k$ events.
$P(X \geq k)$: The cumulative probability of observing $k$ events or more.

Highlighted Results

The calculator highlights cells by their mathematical role. PMF cells show included outcomes, green CDF cells show the endpoint kept in the calculation, red CDF cells show the endpoint subtracted away, and green upper-tail cells show direct $P(X \geq k)$ lookups.

Charts

PMF Chart: A bar graph displaying the probability distribution, where the height of each bar represents $P(X = k)$ for each $k$. This visually shows which outcomes are most likely.
CDF Chart: A stepped graph showing cumulative probability. Each jump lands on the value of $P(X \leq k)$ at an integer $k$, which matches the fact that Poisson outcomes are whole-number counts.

The Poisson distribution is widely applied in various fields to model and analyze the occurrence of independent events over a fixed interval:

Examples

Telecommunications: Modeling the number of calls received by a call center per minute.
Manufacturing: Analyzing the number of defects per unit of product, or per square meter of material.
Healthcare: Predicting the number of patients arriving at an emergency room per hour.
Insurance: Estimating the number of claims submitted to an insurance company per day.
Biology: Counting the number of mutations in a DNA strand per unit length.
Environmental Science: Determining the number of rare species observed in a given area.

When to Use

Independent: The occurrence of one event does not affect the probability of another event occurring.
Constant Average Rate: The mean rate of events ($\lambda$) is consistent throughout the interval.
Rare Events: The probability of an event occurring in a very small sub-interval is proportional to the length of that sub-interval.

What if $\lambda$ (the Poisson mean) is 0, negative, or too large?

The Poisson mean ($\lambda$) represents an average rate and must always be positive. This calculator also caps $\lambda$ at its stable computation limit, so very large values are rejected instead of showing unreliable probabilities.

What if $k$ (the number of events) is negative or a non-integer?

The number of events ($k$) must be a non-negative integer ($k \geq 0$, and whole numbers only). It's impossible to observe a negative or fractional number of events in a count. The calculator will prompt for valid integer inputs.

What if results seem "off"?

If the computed probabilities or statistics don't align with your expectations, double-check that your input for the Poisson mean ($\lambda$) accurately reflects the average rate of events in your real-world scenario. Ensure that the assumptions for the Poisson distribution (independence, constant rate, rarity) are reasonably met for your data.

Poisson Mean ($\mu$ or $\lambda$): The average number of events expected to occur in a fixed interval of time or space. It is the sole parameter of the Poisson distribution.
$k$: The specific number of events of interest whose probability is being calculated. Must be a non-negative integer.
$a$ and $b$: Lower and upper interval bounds. The query symbols ($<$ or $\leq$) show whether each endpoint is excluded or included.
Probability Mass Function (PMF): A function that gives the probability of a discrete random variable taking on a specific value (e.g., exactly $k$ events).
Cumulative Distribution Function (CDF): A function that gives the probability that a random variable is less than or equal to a certain value (e.g., $k$ or fewer events).
Variance ($\sigma^2$): A measure of the spread or dispersion of the distribution. For the Poisson distribution, the variance is equal to its mean ($\lambda$).
Standard Deviation ($\sigma$): The square root of the variance, indicating the typical deviation from the mean in the same units as the mean.

NIST/SEMATECH e-Handbook of Statistical Methods - Poisson Distribution
A technical reference for the PMF, CDF, and distribution properties.
OpenStax Introductory Statistics 2e - Poisson Distribution
A free, peer-reviewed textbook section with accessible examples and practice context.
Penn State University STAT 414/415 - Poisson Distribution
Part of an online course curriculum, this resource offers academic insights into the Poisson distribution within the context of discrete probability distributions.