The equation $P_n(t + h)$ for the Poisson process can be explained by considering the different ways in which $n$ arrivals can occur in the interval $[0, t+h]$, where $t$ and $h$ are non-overlapping sub-intervals. The expression breaks down into several terms that consider the distribution of $n$ arrivals over the two intervals. Let's break it down step-by-step:

1. Definition of $P_n(t + h)$:

   • $P_n(t + h)$ represents the probability of having $n$ arrivals in the combined time interval $[0, t + h]$.

2. Partitioning the Interval:

   • We can partition the interval $[0, t + h]$ into two non-overlapping intervals $[0, t]$ and $[t, t + h]$.

3. Probability of Arrivals in Each Interval:

   • $P_n(t + h)$ is the sum of probabilities of $n$ arrivals occurring in various combinations across $[0, t]$ and $[t, t + h]$.

4. Combination of Arrivals:

   • The equation $P_n(t + h) = P(\text{n arrivals in } t \text{ and none in } h) + P(\text{n-1 arrivals in } t \text{ and 1 in } h) + \ldots + P(\text{no arrivals in } t \text{ and n in } h)$ sums these probabilities.

Let's look at each term:

• $P(n \text{ arrivals in } t \text{ and none in } h)$:

  • The probability of having $n$ arrivals in the interval $[0, t]$ and no arrivals in the interval $[t, t + h]$.
  • This can be written as $P_n(t) \cdot P_0(h)$.

• $P(n-1 \text{ arrivals in } t \text{ and 1 in } h)$:

  • The probability of having $n-1$ arrivals in the interval $[0, t]$ and 1 arrival in the interval $[t, t + h]$.
  • This can be written as $P_{n-1}(t) \cdot P_1(h)$.

• Continuing in the Same Way:

  • We continue this process for all possible combinations of arrivals in the intervals $[0, t]$ and $[t, t + h]$ up to $P(0 \text{ arrivals in } t \text{ and n in } h)$, which can be written as $P_0(t) \cdot P_n(h)$.

5. Summing All Possibilities:
   • The total probability $P_n(t + h)$ is the sum of all these individual probabilities: $P_n(t + h) = P_n(t) P_0(h) + P_{n-1}(t) P_1(h) + P_{n-2}(t) P_2(h) + \cdots + P_0(t) P_n(h)$

Detailed Explanation with Formulas

Using the Poisson distribution properties:

• $P_k(t) = \frac{(\lambda t)^k e^{-\lambda t}}{k!}$
• Where $\lambda$ is the rate of the Poisson process.

Let's rewrite the probabilities: $P_n(t + h) = \sum_{k=0}^{n} P_{n-k}(t) P_k(h)$

Plugging in the Poisson probabilities: $P_n(t + h) = \sum_{k=0}^{n} \frac{(\lambda t)^{n-k} e^{-\lambda t}}{(n-k)!} \cdot \frac{(\lambda h)^k e^{-\lambda h}}{k!}$

Simplifying the sum: $P_n(t + h) = e^{-\lambda (t+h)} \sum_{k=0}^{n} \frac{(\lambda t)^{n-k} (\lambda h)^k}{(n-k)! k!}$

Recognizing the binomial expansion term within the sum: $P_n(t + h) = e^{-\lambda (t+h)} \frac{(\lambda (t + h))^n}{n!}$

Thus, we see that the Poisson distribution for the combined interval $[0, t + h]$ is still Poisson distributed with parameter $\lambda (t + h)$.

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

1. How do we calculate the mean and variance of a Poisson distribution?
2. What are the assumptions underlying the Poisson process?
3. Can you explain the relationship between the Poisson and exponential distributions in more detail?
4. How can we use the Poisson distribution to model rare events in real-world scenarios?
5. What is the probability generating function for the Poisson distribution?

Tip: When working with the Poisson distribution, remember that it is particularly useful for modeling the number of events happening in a fixed period of time or space, especially when these events occur independently and at a constant average rate.