Suppose that {N1(t), t ≥ 0} and {N2(t), t ≥ 0} are independent Poisson processes with rates λ1 and λ2. Show that {N1(t) + N2(t), t ≥ 0} is a Poisson process with rate λ1 + λ2. What is the probability that the first event of the combined process comes from {N1(t), t ≥ 0}?

"To show the superposition of two independent Poisson processes is itself a Poisson process with a rate that is the sum of the two individual rates, we need to verify two key properties of a Poisson process: Independent and Stationary Increments.

Let {N1(t), t ≥ 0} be a Poisson process with rate λ1, and {N2(t), t ≥ 0} with rate λ2.

1. **Independent Increments** property holds because N1(t) and N2(t) are independent processes, implying their superposition has independent increments.
2. **Stationary Increments** property holds as the number of events in any interval for N1(t) and N2(t) only depends on the length of the interval, so this applies to their sum.

Therefore, the combined process {N1(t) + N2(t), t ≥ 0} is a Poisson process.

**Rate of the combined process:**
The rate of the combined process is λ1 + λ2 due to the independence of the processes.

**Probability first event from N1(t):**
The probability is given by: P = \frac{\lambda_1}{\lambda_1 + \lambda_2} "

**Answer:** P = \frac{\lambda_1}{\lambda_1 + \lambda_2}