About Probability Distribution Calculator
This probability distribution calculator is free, needs no sign-up and runs in your browser, so nothing you enter is uploaded. It covers nine distributions: normal, binomial, Poisson, geometric, exponential, uniform, Student t, chi-square and F.
For each one you can compute P(X ≤ x), P(X < x), P(X ≥ x), P(X > x), P(a ≤ X ≤ b) and the density or point probability, and you can run it backwards: give a probability and get the value x with that much probability below it. The mean, variance and standard deviation are shown, and a chart shades the region you asked about.
Continuous distributions are drawn as smooth curves with the tail or interval shaded. Discrete ones are drawn as bars, with the bars inside the region highlighted. Every answer comes with a short worked step and the formula of the distribution in KaTeX.
How to use Probability 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.
PDF, PMF and CDF
A continuous distribution has a density f(x), and probabilities are areas under it: P(a ≤ X ≤ b) = F(b) - F(a), where F is the cumulative distribution function. A discrete distribution has a mass function P(X = k), and F(k) is the running total of those masses.
The inverse of F is the quantile function. It answers questions like "which score cuts off the top 5%". Each distribution page shows its own density or mass function.
F(x) = P(X ≤ x)
P(X > x) = 1 − F(x)
P(a ≤ X ≤ b) = F(b) − F(a) (continuous)
P(a ≤ X ≤ b) = F(b) − F(a − 1) (integer values)Worked examples
Normal, mean 100, standard deviation 15: P(85 ≤ X ≤ 115) gives 0.68269
This is the share of values within one standard deviation of the mean, about 68%.
Poisson with λ = 3: P(X = 2) gives 0.22404
e^-3 × 3² ÷ 2! = 0.2240.
Binomial with n = 10, p = 0.5: P(X ≤ 3) gives 0.17188
The masses for 0, 1, 2 and 3 successes add up to 176 ÷ 1024.
Which distribution fits
| Distribution | Type | Use it for |
|---|---|---|
| Normal | Continuous | Measurements that cluster around an average |
| Binomial | Discrete | Successes in a fixed number of yes or no trials |
| Poisson | Discrete | Counts of events in a fixed interval |
| Geometric | Discrete | Trials until the first success |
| Exponential | Continuous | Waiting time between events |
| Uniform | Continuous | Every value in a range equally likely |
| Student t | Continuous | Means with an estimated standard deviation |
| Chi-square | Continuous | Variances and goodness of fit |
| F | Continuous | Ratios of variances, ANOVA |
Good to know
Limits of this calculator
Binomial trials are limited to 100000 and the Poisson rate to 100000. A discrete x must be a whole number. Mean and variance are shown as undefined when they do not exist, for example the mean of a t distribution with 1 degree of freedom. Values come from numerical algorithms accurate to many digits, but extreme tails beyond about 1e-300 are not meaningful.
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.
What is the difference between a PDF and a CDF?
The PDF (or PMF for discrete variables) gives the density or the probability of one value. The CDF gives the probability of being at or below a value, so it adds up the PDF from the left.
How do I find the probability between two values?
Choose P(a ≤ X ≤ b), then enter a and b. The calculator subtracts the CDF at a from the CDF at b.
What is the inverse or quantile?
It is the value x such that a given probability lies at or below it. The 0.975 quantile of the standard normal is 1.96.
Why does a discrete x need to be a whole number?
Binomial, Poisson and geometric variables count things, so only whole numbers have probability. Enter an integer.
Why is the mean shown as undefined?
Some heavy-tailed cases have no mean or variance. A t distribution needs more than 1 degree of freedom for a mean and more than 2 for a variance.
Popular Probability Distribution Calculator pages
- 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.
- Poisson Distribution CalculatorEnter the average rate λ to get the chance of exactly k events, at most k, at least k or a range.
- 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.