About Poisson Distribution Calculator
This Poisson distribution calculator gives the probability of a number of events in a fixed interval when events happen independently at a constant average rate λ. It opens on λ = 3 and finds P(X = 2). It is free, needs no sign-up and runs in your browser, so nothing you enter is uploaded.
Typical uses are calls per hour, typos per page, or accidents per week. The distribution has one parameter, and its mean and variance are both equal to λ, so a count whose variance is much larger than its mean is not Poisson.
If the rate is for a different length of time, scale it first. An average of 6 calls per hour is λ = 3 for a 30 minute window.
How to use Poisson Distribution Calculator
- Choose the distribution from the list. Each has its own parameters, such as the mean and standard deviation for the normal.
- Enter the parameters. The hint under each field says what is allowed.
- Choose what to calculate: a lower tail, an upper tail, an interval, a density or point probability, or the inverse.
- Enter x, or a and b for an interval, or the probability for the inverse. The result updates as you type.
- Read the answer, the mean and variance, and the shaded chart. Open the working to see which CDF values were used.
- Copy the result with the copy button.
Features
- Nine distributions: normal, binomial, Poisson, geometric, exponential, uniform, Student t, chi-square and F.
- Lower and upper tails, strict and non-strict, and intervals P(a ≤ X ≤ b).
- Density for continuous and point probability for discrete distributions.
- Inverse (quantile) calculation, including critical values for t, chi-square and F.
- Mean, variance and standard deviation, shown as undefined where they do not exist.
- Bell curve with shaded tail or interval for continuous distributions, and a bar chart for discrete ones.
- Upper tails computed directly, so very small p-values keep their precision.
- Formula of the distribution in KaTeX and a short worked step for every answer.
Poisson probability mass function
The chance of exactly k events depends only on λ and k.
P(X = k) = λ^k × e^(−λ) / k!
mean = λ
variance = λWorked examples
λ = 3: P(X = 2) gives 0.22404
e^-3 × 3² ÷ 2 = 0.2240.
λ = 4 calls per hour: P(X ≥ 7) gives 0.11067
The chance of seven or more calls in an hour at an average of four.
Privacy: how it works
- Everything is calculated by code running in your browser tab. Nothing you enter is sent to Ulkaa or to any other server, and there is nothing to sign in to.
Tips
- For a discrete distribution, P(X < 3) and P(X ≤ 3) are different. For a continuous one they are the same.
- For a two-sided critical value, such as the 95% value of t, put the inverse probability at 0.975.
- A density can be above 1 when the spread is small. It is not a probability.
- If your question is "at least k", use P(X ≥ x), not P(X > x).
Frequently asked questions
Is it really free?
Yes. There is no price, no trial and no credit card. Ulkaa Tool does the work with your own device, so there is no server bill to pass on to you.
Do I need an account?
No. There is no sign-up and no login. Open the page and use it. An Ulkaa account is not needed for anything on this tool.
Does it run in my browser?
Yes. Your numbers are processed by code running in your browser tab, on your own device.
Is my data uploaded?
No. Your numbers are never sent to Ulkaa or to anyone else. You can check this yourself by watching the Network tab in your browser developer tools while you use the tool.
When should I use a Poisson distribution?
For counts of independent events in a fixed interval of time or space when the average rate is steady, such as arrivals, defects or decays.
How do I change the time window?
Multiply the rate by the length of the window. A rate of 2 per day is λ = 14 for a week.
How is Poisson related to the binomial?
With many trials and a small success chance, a binomial is close to a Poisson with λ = n × p.
More Probability Distribution Calculator pages
- Probability Distribution CalculatorPick a distribution, set its parameters and get PDF, CDF, interval probabilities, quantiles, mean and variance with a chart.
- Normal Distribution CalculatorEnter a mean and standard deviation to get tail probabilities, intervals and percentiles on a bell curve.
- Binomial Distribution CalculatorEnter the number of trials and the chance of success to get P(X = k), P(X ≤ k), P(X ≥ k) and ranges.
- Geometric Distribution CalculatorEnter the success probability to get the chance that the first success happens on or by trial k.
- Exponential Distribution CalculatorEnter the rate λ to get the chance a waiting time exceeds, or stays below, a value, and its percentiles.
- Uniform Distribution CalculatorEnter the minimum a and maximum b to get the chance of landing in any sub-interval.
- Student t Distribution CalculatorEnter the degrees of freedom to get t critical values, tail areas and p-values on a shaded t curve.
- Chi-Square Distribution CalculatorEnter the degrees of freedom to get chi-square p-values, critical values and tail areas.
- F Distribution CalculatorEnter both degrees of freedom to get F p-values, critical values and tail areas.