Exponential distribution

The exponential distribution models the waiting time until the next of a series of independent events that occur at a constant average rate λ, with mean waiting time 1 ÷ λ.

ByHoang TruongUpdated July 5, 2026

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A support line receives calls at an average rate of 12 an hour, or λ = 0.2 per minute. The probability that more than 10 minutes passes before the next call is e^(−0.2 × 10) = e^(−2) ≈ 13.5%. The mean waiting time is 1 ÷ 0.2, or 5 minutes, and because the process is memoryless, that 13.5% figure holds no matter how long the wait has already lasted.

Where it fits

TopicProbability & DistributionsAdvancedSubjectData Analysis & StatisticsAdvanced

The formula

LaTeX

P(T > t) = e^{-\lambda t}

Variables

$T$: Waiting time until the next event
$t$: A specific time value being tested
$λ$: Average rate of events per unit of time (must be positive)
$e$: Euler's number, approximately 2.71828

Probability that the waiting time T until the next event exceeds t, given events occur independently at constant average rate λ.

LaTeX

E[T] = \frac{1}{\lambda}

Variables

$E [T]$: Expected (mean) waiting time until the next event
$λ$: Average rate of events per unit of time

The average time between events is the reciprocal of the event rate; a higher rate of events means a shorter average wait.