Exponential and Uniform Distribution Calculator
Find probabilities, the mean and the variance for the exponential and uniform distributions, with the area shaded.
About the Exponential and Uniform Distribution Calculator
The exponential distribution models the waiting time until the next event when events happen at a constant rate λ: the time to the next call, or the life of a component that doesn't wear out. P(X < x) = 1 − e^(−λx), with mean 1/λ. It is memoryless: having waited s already doesn't change the chance of waiting t more.
The uniform distribution on [α, β] makes every value equally likely, so probabilities are lengths: P(a < X < b) = (b − a)/(β − α), with mean (α + β)/2 and variance (β − α)²/12.
The formula
Exponential: f(x) = λe^(−λx), F(x) = 1 − e^(−λx); uniform: f(x) = 1/(β − α)
- Exponential mean, variance
- 1/λ, 1/λ²
- Uniform mean, variance
- (α + β)/2, (β − α)²/12
Worked example
Calls arrive at λ = 0.5 per minute
- P(wait < 2 min) = 1 − e^(−1)
- = 0.6321
Questions people ask
How is it linked to the Poisson distribution?
If events follow a Poisson process with rate λ, the gaps between them are exponential with the same λ.
What does memoryless mean?
A component that has already lasted 100 hours has the same chance of lasting another 50 as a new one.