What Is Exponential Probability?
The exponential distribution models the time until the next event when events occur independently at a constant average rate λ. Its cumulative probability is P(X ≤ x) = 1 − e^(−λx), and its mean waiting time is 1/λ.
Enter either the rate or the mean, choose whether you want a lower tail, upper tail or range probability, and the calculator gives the probability together with the median ln(2)/λ, variance, standard deviation and percentiles. Two charts show the density and the cumulative probability.
The article below explains the formula, the variables, the memoryless property and a worked example, and answers common questions about rate versus mean and Poisson versus exponential.
How to Use the Exponential Probability Calculator
- Choose whether you know the rate λ or the mean waiting time.
- Enter that number.
- Choose ≤, > or between and enter a (and b).
- Read the probability, mean, median, variance and charts.
Exponential Probability Formula
Density
Cumulative probability
Mean, variance and median
Exponential Probability Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| λ | Rate parameter | events per unit time |
| 1/λ | Mean waiting time | |
| x | Waiting time | must be 0 or more |
| F(x) | Cumulative probability | 1 − e^(−λx) |
Exponential Probability Worked Example
A call centre receives a call on average every 10 minutes, so λ = 0.1 per minute. The chance the next call arrives within 5 minutes is P(X ≤ 5) = 1 − e^(−0.1 × 5) = 1 − 0.6065 = 0.3935, about 39.4%. The chance of waiting more than 15 minutes is e^(−1.5) = 0.2231.
With the example values loaded in the calculator above, the steps are:
- Rate λ = 1 ÷ mean = 1 ÷ 10 = 0.1; mean = 1 ÷ λ = 10.
- P(X ≤ x) = 1 − e^(−λx) = 1 − e^(−0.1 × 5) = 0.39346934.
Result: P(X ≤ 5) = 0.39346934; Mean 1/λ = 10.
The memoryless property
If you have already waited s minutes, the chance of waiting at least t more is the same as it was at the start: P(X > s + t | X > s) = P(X > t). The distribution has no memory of the past.
Where Exponential Probability Is Used
Queues and call centres
Estimating the chance the next call arrives within a given time.
Reliability
Modelling the time to failure when the failure rate is constant.
Radioactive decay
Describing the time until the next decay.
Probability coursework
Checking CDF, mean and median answers.
Common Exponential Probability Mistakes to Avoid
- Keep the rate and the x value in the same time unit.
- A mean of 10 minutes is a rate of 0.1 per minute.
- The standard deviation equals the mean.
Exponential Probability Terms and Definitions
- Rate λ
- The average number of events per unit of time.
- Poisson process
- Events occurring independently at a constant average rate; waiting times between events are exponential.
Exponential Probability FAQs
How do I calculate exponential probability?
P(X ≤ x) = 1 − e^(−λx) and P(X > x) = e^(−λx), where λ is the rate.
What is the relationship between rate and mean?
Mean = 1/λ, so a rate of 0.2 per hour means a mean wait of 5 hours.
Why is the median smaller than the mean?
The distribution is right-skewed; the median is ln(2)/λ ≈ 0.693 times the mean.
What is the difference between exponential and Poisson?
Poisson counts events in a fixed period; exponential measures the time between events.