Exponential distribution
Undergraduate
Consider events that happen independently of each other at an average rate of $\lambda$ per unit of time, such as customers arriving at a shop. The waiting time $T$ until the first event has an exponential distribution with rate $\lambda$, which has the probability density
\[f(t) = \lambda e^{-\lambda t} \quad \text{for } t \geq 0.\]Examples
Worked examples
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Customers arriving at a shop independently of each other at an average rate of $3$ per hour
: The waiting time, in hours, until the first customer: exponential with rate $3$
Undergraduate
For customers arriving independently at an average rate of $3$ per hour, the waiting time until the first customer has an exponential distribution with rate $3$, so the expected wait is $\frac{1}{3}$ hour, or $20$ minutes.
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Emails arriving independently of each other at an average rate of $6$ per hour, and the event that no email arrives in the next $10$ minutes
: The waiting time $T$, in minutes, until the next email is exponential with rate $\frac{1}{10}$ per minute, so $P(T > 10) = e^{-1} \approx 0.368$
Undergraduate
For emails arriving at $6$ per hour, the waiting time until the next email is a continuous time, so the waiting time has an exponential distribution with mean $10$ minutes.
Properties
- An exponential random variable with rate $\lambda$ has mean $1 / \lambda$ and satisfies $P(T > t) = e^{-\lambda t}$.
- The exponential distribution is memoryless: $P(T > s + t \mid T > s) = P(T > t)$. The exponential distribution is the continuous counterpart of the geometric distribution.
Distinctions involving this term
- Poisson vs. exponential distribution: Consider events that happen independently of each other at a constant average rate $\lambda$, such as customers arriving at a shop (a Poisson process). The Poisson distribution describes the number of events in a fixed length of time, which is a whole number. The exponential distribution describes the time until the next event, which is a positive real number.