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.
How to Read an Exponential Probability
The exponential distribution models the waiting time until the next event when events occur at a constant average rate. If a call arrives on average every 10 minutes, λ = 0.1 per minute, and the chance the next call arrives within 5 minutes is 1 − e^(−0.5) = 0.3935.
The mean wait is 1 ÷ λ = 10 minutes, but the median is ln 2 ÷ λ = 6.93 minutes, which is smaller because the distribution is skewed to the right. A property called memorylessness means the time already waited does not change the chance of the next event arriving soon.
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
Waiting times
Find the chance that the next arrival comes within a given time.
Lifetimes
Model the lifetime of a part that fails at a constant rate.
Percentiles
See the time by which 50%, 90% or 99% of events have occurred.
Survival
Find the chance of waiting longer than a given time.
Common Exponential Probability Mistakes to Avoid
- Rate and mean are reciprocals. Make sure you enter the one you know in the right box.
- Use the same time unit for the rate and for x.
- The model assumes a constant rate. A part that wears out does not fit.
- For counts in a fixed interval, use the Poisson distribution instead.
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?
Use P(X ≤ x) = 1 − e^(−λx). For λ = 0.1 and x = 5 that is 1 − e^(−0.5) = 0.3935.
What is the relationship between rate and mean?
The mean is 1 ÷ λ. A rate of 0.1 per minute means an average wait of 10 minutes.
Why is the median smaller than the mean?
The distribution has a long right tail, which pulls the mean up. The median is ln 2 ÷ λ, about 69% of the mean.
What is the difference between exponential and Poisson?
The exponential gives the waiting time between events. The Poisson gives the number of events in a fixed interval. They describe the same process from two sides.