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SigmaCalc
The distribution behind 'how long until the next event' — customer arrivals, equipment failures, radioactive decay.
The exponential distribution has a distinctive quirk: the probability of waiting an additional x time units doesn't depend on how long you've already waited. A machine that hasn't failed in the first hour is exactly as likely to fail in the next hour as a brand-new one.
λ (lambda) must be expressed in the same time units as x — if your average event rate is per hour but you're asking about a probability over 30 minutes, convert one of them to match the other first.
P(X ≤ x)
0.632
F(x) = 1 − e^(−λx)
What you entered
Probability density at x
0.5 × e^(−0.5 × 2)= 0.1839P(X ≤ x) — cumulative probability
1 − e^(−0.5 × 2)= 0.6321Result
P(X ≤ x): 0.632
At a rate of 0.5 events per unit time, there's a 63.21% chance of waiting 2 or less time units for the next event — the exponential distribution models memoryless waiting times.