Exponential Distribution Calculator
Compute P(X ≤ x), P(X > x) and P(a < X ≤ b) from either the rate λ or the mean 1/λ — with a shaded PDF diagram and full step-by-step working.
The exponential distribution explained, step by step
The exponential distribution models waiting time between events in a Poisson process — same rate λ as the Poisson Distribution Calculator, different question ("how long between events?" instead of "how many in an interval?"). Each card below covers one piece the calculator uses.
1. PDF and CDF — the shape and the cumulative area
The exponential PDF starts at height λ at x = 0 and decays exponentially. The CDF integrates that curve to give the probability of waiting at most x units of time — it climbs quickly at first and asymptotes to 1.
2. Survival and interval probabilities
The survival function is the complement of the CDF — the chance you wait longer than x. Any interval probability is the difference of two survivals (or two CDFs), which is why the shaded area between a and b uses e^(−λa) − e^(−λb).
3. Mean = SD = 1/λ (signature property)
Both the mean and the standard deviation equal 1/λ, and the variance is 1/λ². The median is smaller — ln(2)/λ — because the distribution is heavily right-skewed and the long tail pulls the mean above the median.
4. Memoryless — no aging in the distribution
Exponential is the only continuous distribution with the memoryless property: given you've already waited s units, the chance of waiting at least t more is the same as if you'd just started. This is why exponential fits Poisson-process waits (calls, decays) but not wear-out lifetimes — for those use a Weibull.
Features of this calculator
- Enter either the rate λ or the mean 1/λ — the other value is derived automatically so you don't have to convert by hand.
- Supports three probability queries: P(X ≤ x), P(X > x), and P(a < X ≤ b) between two values.
- Reports mean, variance and standard deviation with a reminder that mean = SD = 1/λ.
- Shaded PDF curve shows exactly which region of the exponential distribution is being computed.
- Show/hide step-by-step working writes out F(x) = 1 − e^(−λx) with your numbers substituted in.
Frequently asked questions
+How do I estimate λ from data?
The maximum-likelihood estimate of λ is 1 divided by the sample mean of the observed waiting times. So if the average time between calls in your data is 7 minutes, an estimate of λ is 1/7 ≈ 0.143 per minute.
+Why isn't the mean the same as the median?
The exponential distribution is heavily right-skewed. The median is ln(2)/λ ≈ 0.693/λ, always noticeably less than the mean 1/λ. Half of the wait times fall below the median, but the long tail on the right pulls the mean up.
+Is exponential the same as exponential decay in physics?
Very closely related. Radioactive decay is a canonical Poisson process, and the waiting time until any single atom decays is exponentially distributed. The macroscopic decay law N(t) = N₀·e^(−λt) is the survival function of a huge number of independent exponential lifetimes.
+How is exponential related to the geometric distribution?
The geometric distribution is the discrete analogue — waiting time (in whole trials) for the first success in a sequence of Bernoulli trials. Exponential is its continuous counterpart, and it inherits the memoryless property from geometric.
+What if my data has an obviously increasing failure rate?
Then exponential is the wrong model. Look at the Weibull distribution, which generalises exponential with a shape parameter that lets the hazard rate increase (wear-out), decrease (infant mortality) or stay constant (exponential is Weibull with shape = 1).
+Can I use this for the time between website visits?
As a first approximation during a stable traffic window, yes. Beware daily and weekly cycles — visits between 3am and 4am aren't drawn from the same distribution as visits between 2pm and 3pm, so fit exponential inside a homogeneous window rather than across a full day.