About Exponential Distribution Calculator
This exponential distribution calculator models the waiting time until the next event in a process with a constant rate λ. It opens on λ = 0.5 per unit of time and finds the chance of waiting more than 3. It is free, needs no sign-up and runs in your browser, so nothing you enter is uploaded.
The exponential is the continuous partner of the geometric distribution and is also memoryless. Its mean is 1 ÷ λ, so a rate of 0.5 events per minute means an average wait of 2 minutes.
The survival function has a simple closed form, P(X > x) = e^(-λx), which the calculator uses directly for upper tails. The inverse gives percentiles, for example the median wait is ln 2 ÷ λ.
How to use Exponential 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.
Exponential distribution
The density falls off steadily from its peak at zero.
f(x) = λ × e^(−λx), x ≥ 0
F(x) = 1 − e^(−λx)
mean = 1 / λ
variance = 1 / λ²Worked examples
λ = 0.5: P(X > 3) gives 0.22313
e^-1.5 = 0.2231, the chance of waiting more than 3 units.
λ = 0.5: median waiting time gives 1.3863
ln 2 ÷ 0.5 = 1.386, less than the mean of 2 because the distribution is skewed.
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.
Is the exponential rate or the mean the parameter?
This calculator takes the rate λ. If you know the mean wait m, enter λ = 1 ÷ m.
Why is the median smaller than the mean?
The distribution is right-skewed. Many waits are short and a few are very long, which pulls the mean above the median.
How is it related to the Poisson distribution?
If events arrive as a Poisson process, the counts per interval are Poisson and the waits between events are exponential.
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.
- 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.
- 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.