The Poisson distribution can be derived using calculus from the concept of a Poisson process. A Poisson process is a stochastic process that models the occurrence of events that happen independently of each other at a constant average rate. Here’s the detailed proof:

1. Definition of a Poisson Process:

   • Let $N(t)$ be the number of events that occur in the interval $[0, t]$.
   • The events occur independently.
   • The probability of a single event occurring in a small interval $[t, t + \Delta t]$ is proportional to $\Delta t$, i.e., $P(\text{event in } [t, t + \Delta t]) = \lambda \Delta t + o(\Delta t)$, where $\lambda$ is the average rate of events per unit time and $o(\Delta t)$ denotes higher-order small terms.

2. Derivation of the Poisson Distribution:

   • We need to find the probability $P(N(t) = k)$ that exactly $k$ events occur in the interval $[0, t]$.
   • Consider dividing the interval $[0, t]$ into $n$ sub-intervals of equal length $\Delta t = \frac{t}{n}$.

3. Limiting Process:

   • As $n$ becomes large, $\Delta t$ becomes small, and we assume the events occurring in each sub-interval are independent.
   • For large $n$, the probability of more than one event in a small interval $\Delta t$ becomes negligible, so the probability of exactly one event in $\Delta t$ is $\lambda \Delta t$, and the probability of no event is $1 - \lambda \Delta t$.

4. Probability Calculation:

   • The probability of exactly $k$ events in $n$ sub-intervals (each of length $\Delta t$) is given by a binomial distribution: $P(N(t) = k) = \binom{n}{k} (\lambda \Delta t)^k (1 - \lambda \Delta t)^{n-k}$
   • Substitute $\Delta t = \frac{t}{n}$: $P(N(t) = k) = \binom{n}{k} \left(\lambda \frac{t}{n}\right)^k \left(1 - \lambda \frac{t}{n}\right)^{n-k}$

5. Taking the Limit:

   • As $n \to \infty$, $\binom{n}{k}$ can be approximated using Stirling's approximation: $\binom{n}{k} \approx \frac{n^k}{k!}$
   • Therefore, $P(N(t) = k) \approx \frac{n^k}{k!} \left(\lambda \frac{t}{n}\right)^k \left(1 - \frac{\lambda t}{n}\right)^{n-k}$
   • Simplify the expression: $P(N(t) = k) \approx \frac{(\lambda t)^k}{k!} \left(1 - \frac{\lambda t}{n}\right)^n \left(1 - \frac{\lambda t}{n}\right)^{-k}$

6. Exponential Limit:

   • As $n \to \infty$, $\left(1 - \frac{\lambda t}{n}\right)^n \to e^{-\lambda t}$
   • Hence, $P(N(t) = k) \approx \frac{(\lambda t)^k}{k!} e^{-\lambda t}$

Thus, we derive the Poisson distribution: $P(N(t) = k) = \frac{(\lambda t)^k}{k!} e^{-\lambda t}$

This is the probability of observing $k$ events in a Poisson process with rate $\lambda$ over a time interval $t$.

Do you want details or have any questions? Here are five questions you might ask next:

1. What are some real-world examples of Poisson distribution applications?
2. Can you explain the relationship between the Poisson and exponential distributions?
3. How can we use the Poisson distribution to model rare events?
4. What are the mean and variance of a Poisson distribution?
5. How does the Poisson distribution relate to the binomial distribution?

Tip: When working with the Poisson distribution, remember that it is especially useful for modeling the number of events occurring within a fixed interval of time or space, particularly when the events are rare and independent.