Example · Undergraduate
Terms
| Term | This example | Why |
|---|
| uniform distribution |
If exactly one customer arrives between 1:00 and 2:00, then the arrival time: uniform on that hour
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For customers arriving independently at a constant rate, if exactly one customer arrives in a given hour, then the customer is equally likely to arrive at each time in that hour, so the arrival time has a uniform distribution on the hour.
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| Bernoulli distribution |
Whether at least one customer arrives in an hour: Bernoulli with $p = 1 - e^{-3} \approx 0.950$
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For customers arriving at an average rate of $3$ per hour, the indicator that at least one customer arrives in an hour has a Bernoulli distribution with success probability $1 - P(\text{no customers}) = 1 - e^{-3}$.
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| Poisson distribution |
The number of customers in an hour: Poisson with mean $3$
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For customers arriving independently at an average rate of $3$ per hour, the number of customers in an hour has a Poisson distribution with mean $3$, because the number counts independent events at a constant rate.
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| exponential distribution |
The waiting time, in hours, until the first customer: exponential with rate $3$
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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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| normal distribution |
The number of customers in $100$ hours: approximately $\mathcal{N}(300, 300)$
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For customers arriving at an average rate of $3$ per hour, the number of customers in $100$ hours is Poisson with mean $300$, which is approximately normal with mean $300$ and variance $300$ by the central limit theorem.
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Distinctions using this example
- Poisson vs. exponential distribution: For customers arriving at $3$ per hour, the Poisson distribution gives the number of customers in an hour, with mean $3$, and the exponential distribution gives the wait until the first customer, with mean $\frac{1}{3}$ hour.
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