Answer First
We use the Poisson distribution to model the number of events in a time interval, and we use the Exponential distribution to model the waiting time between events. They describe different aspects of the same arrival process: Poisson counts events, while Exponential measures the time until the next event.
Problem Setup
The Poisson distribution models counts:
\[ X \sim Pois(\lambda) \]
The Exponential distribution models waiting times:
\[ T \sim Exp(\lambda) \]
The key link is:
\[ X \sim Pois(\lambda t) \quad \Longleftrightarrow \quad T \sim Exp(\lambda) \]
Step-by-Step Solution
1. Poisson answers “How many events occur?”
Examples:
- How many customers arrive in 10 minutes?
- How many emails arrive per hour?
- How many machine failures occur per month?
2. Exponential answers “How long until the next event?”
Examples:
- How long until the next customer arrives?
- How long until the next website click?
- How long until the next machine failure?
3. The Poisson and Exponential describe the same process
If arrivals follow a Poisson process with rate λ:
- The number
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