How this instrument works
The exponential distribution describes the waiting time until a single random event occurs, given that such events happen at a steady average rate λ (lambda) — the same rate that governs a Poisson process. If phone calls arrive at a call center at an average rate of λ = 2 per minute, the time until the very next call follows an exponential distribution with that same λ. It is a continuous distribution defined only for x ≥ 0, since a waiting time can never be negative.
Two functions describe it. The probability density function, f(x) = λ × e^(−λx), does not give a probability directly (as with any continuous distribution, that requires an interval), but it shapes the distribution: it is tallest near x = 0 and decays smoothly toward zero as x grows, reflecting that short waits are more common than long ones. The cumulative distribution function, F(x) = 1 − e^(−λx), gives the actual probability that the wait is x or less — a genuinely useful number you can quote directly.
The exponential distribution's defining trait is memorylessness: the probability of waiting an additional t units, given that you have already waited some amount of time, is exactly the same as the probability of waiting t units from scratch. Past waiting time gives you no information about how much longer you'll wait — a component that hasn't failed yet is, under this model, no more or less likely to fail in the next minute than a brand-new one. It is a strong assumption, appropriate for random arrivals and certain electronic-component lifetimes, but not for wear-out failures where age genuinely matters (the Weibull distribution's shape parameter is built to capture that instead).
- Enter the average event rate into Rate parameter (λ) — events per unit time or space, must be greater than zero.
- Enter the waiting time you want to evaluate into x (value, x ≥ 0) — the same time units as λ.
- Read Probability density f(x) — the density of the distribution at that x, used for shape and comparison, not a direct probability.
- Read Cumulative probability F(x) = P(X ≤ x) — the actual probability that the wait is x or shorter, a number between 0 and 1.
- To find the probability of waiting longer than x instead, compute 1 − F(x), since F(x) already covers 'x or less'.
Worked example — waiting time with rate λ = 2, evaluated at x = 0.5
Enter λ = 2 into Rate parameter and x = 0.5 into x — a rate of 2 events per minute, evaluated at a wait of half a minute. The instrument first computes λ × x = 2 × 0.5 = 1, then e^(−1) ≈ 0.36787944.
Probability density f(x) reads 0.735758882343, computed as λ × e^(−λx) = 2 × 0.36787944. Cumulative probability F(x) reads 0.632120558829, computed as 1 − 0.36787944. That second number is the one to quote directly: there is about a 63.2% chance the next event arrives within half a minute, and correspondingly about a 36.8% chance it takes longer than that.
Questions
What is the difference between the exponential PDF and CDF?
The cumulative distribution function F(x) is the one to quote as an actual probability: it gives P(X ≤ x), the chance the wait is x or less, always between 0 and 1. The probability density function f(x) is not itself a probability for a continuous distribution — it describes the relative likelihood at each point and is used to compute probabilities over intervals (by integration) or to compare how the distribution's shape changes with λ.
What does 'memoryless' mean for the exponential distribution?
It means the distribution has no memory of elapsed time: if you have already waited 5 minutes without an event, the probability of waiting an additional 2 minutes is exactly the same as the probability of waiting 2 minutes from a fresh start. Among all continuous distributions, the exponential distribution is the only one with this property, which makes it the natural model for constant-rate random arrivals and, in reliability engineering, for failures that strike without warning rather than from gradual wear.
How is the exponential distribution related to the Poisson distribution?
They describe the same underlying random process from two different angles. If events occur at a steady average rate λ, the Poisson distribution (this site's Poisson Distribution calculator) gives the probability of a specific count of events in a fixed interval, while the exponential distribution gives the probability distribution of the waiting time between consecutive events. Both are governed by the same rate parameter λ.
How does this relate to the Weibull distribution?
The exponential distribution is the special case of the Weibull distribution when the shape parameter k = 1 and the Weibull scale parameter equals 1/λ. This site's Weibull Distribution calculator confirms it directly: entering k = 1, λ = 2, x = 2 reduces to a PDF of 0.183939720586 and a CDF of 0.632120558829 — exactly the same PDF and CDF values as an exponential distribution with rate 1/2 evaluated at x = 2. The Weibull distribution's extra shape parameter lets it model failure rates that increase or decrease with age, something the memoryless exponential distribution cannot do.
What is the mean of an exponential distribution?
The mean (expected waiting time) is 1/λ. For the worked example with λ = 2 events per minute, the mean wait is 1/2 = 0.5 minutes — which is also why F(0.5) works out to just over 63%, since the cumulative probability at the mean of an exponential distribution is always 1 − e^(−1) ≈ 63.2%, regardless of the specific value of λ.
Can λ or x be negative or zero?
λ must be strictly greater than zero — a rate of zero or fewer events per unit time makes the distribution undefined, since events would never occur. x, the waiting time, can be zero (the probability of waiting no time at all is captured by F(0) = 0) but cannot be negative, since a negative wait has no physical meaning.