In a network of packages there are two transmission lines.One has λ=10 packages/s and Number Two has λ=20p/s.In an interval of 5 seconds 100 packages arrived.
So the first question is whats the probability of 40p coming from line 1 and 60 from line 2. I figured it would be
Pt=(P(5)12=40)+(P(5)22=60)
And the second problem is the calculate the probability of two consecutive packages being separated by less than 10 ms.
I Have no idea how to solve this once since the lines overlap right?Should I calculate a poisson distribution for two consecutive packages being one from line 1 and another from line 2, and another one with 2 from line 2? Thank you for your time
Each packet has a $10/(10+20)=1/3$ chance of being from line $1$. So the probability that of the first $100$ packets, $40$ of them are from line $1$ is given by a binomial distribution: $$ \binom{100}{40}(1/3)^{40}(1-1/3)^{100-40} \approx 0.03075091. $$
For the second question, recall that the superposition of two Poisson processes is again a Poisson process with its rate the sum of the rates of the constituent processes. So we may consider a single Poisson process with rate $\lambda = 30$ (per second). The time between consecutive arrivals is exponentially distributed with the same rate $\lambda$, so the probability that two arrivals are separated by less than $1/100$ seconds is given by
$$ 1-\exp(30\cdot 1/100) \approx 0.1845155. $$