Example · Undergraduate
Terms
| Term | This example | Why |
|---|
| Poisson distribution |
The number $N$ of emails in $10$ minutes is Poisson with mean $1$, so $P(N = 0) = e^{-1} \approx 0.368$
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For emails arriving at $6$ per hour, the number of emails in $10$ minutes is a count in a fixed time, so the number has a Poisson distribution with mean $6 \cdot \frac{10}{60} = 1$.
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| exponential distribution |
The waiting time $T$, in minutes, until the next email is exponential with rate $\frac{1}{10}$ per minute, so $P(T > 10) = e^{-1} \approx 0.368$
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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.
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Distinctions using this example
- Poisson vs. exponential distribution: For emails arriving at $6$ per hour, “no email in the next $10$ minutes” is the same event as “zero emails in $10$ minutes” and “the wait is longer than $10$ minutes,” so the Poisson and exponential distributions give the same probability $e^{-1}$.
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