Exponential Distribution Calculator
This exponential distribution calculator needs the rate λ and a limit x. Example: with a mean waiting time of 2 minutes (λ = 0.5), P(X ≤ 3) = 0.77687.
Example: Rate λ 0.5 · Limit x 3 · Find: P(X ≤ x): event by time x → Result: P(X ≤ 3) = 0.77687 (77.687%), λ = 0.5. Source: Wikipedia – Exponential distribution. Updated: .
How it is calculated
The exponential distribution describes the waiting time until the next event when events occur independently at a constant rate λ, such as calls, failures or arrivals.
Formulas
The density is f(x) = λ · e^(−λx) for x ≥ 0. The cumulative distribution is F(x) = P(X ≤ x) = 1 − e^(−λx), and the complement is P(X > x) = e^(−λx). For an interval, P(a < X ≤ b) = e^(−λa) − e^(−λb). Mean and standard deviation both equal 1/λ, the variance is 1/λ² and the median is ln 2 / λ.
Worked example
A hotline takes a call every 2 minutes on average, so λ = 0.5 per minute. How likely is the next call within 3 minutes? e^(−1.5) = 0.22313, so P(X ≤ 3) = 1 − 0.22313 = 0.77687, which is 77.687%. The density at x = 3 is 0.5 · 0.22313 = 0.111565. The median is ln 2 / 0.5 = 1.38629 minutes, below the mean of 2, because the distribution is right-skewed.
- The distribution is memoryless: P(X > s + t | X > s) = P(X > t).
- If λ is a rate per unit of time, x and λ must use the same unit.
- Counting events in a fixed period gives the Poisson distribution with the same λ.
Frequently asked questions
How do you calculate the exponential distribution?
Use F(x) = 1 − e^(−λx). For λ = 0.5 and x = 3 that is 1 − e^(−1.5) = 0.77687.
What is the difference between the rate λ and the mean?
The mean is 1/λ. If an event comes every 2 minutes on average, the rate is λ = 0.5 per minute.
Why is the median below the mean?
The distribution is strongly right-skewed: a few very long waits pull the mean up, while the median ln 2 / λ ≈ 0.693 / λ stays smaller.
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