# Poisson processes of two airline companies. (two independent Poisson processes)

Easyjet and KLM planes request landing permission at Heathrow airport according to independent Poisson processes with intensities $$\lambda$$ and $$\mu$$ per hour, respectively for Easyjet and KLM.

a) Given that exactly three Easyjet planes have requested landing permission over the course of a two-hour period, what is the probability that at least one of these requests occurred during the first hour of that period?

This is a conditional event. I am having serious trouble thinking how to write this down however... which makes me feel rather stupid.

b) What is the probability that among the first three requests for landing permission exactly two are generated by KLM planes?

Two of the events should be KLM events, the other one is an Easyjet event, this probability is given by: $$\binom {3}{2} \frac{\lambda}{\lambda+ \mu} \cdot \left( \frac{\mu}{\lambda + \mu} \right)^2.$$

The first one is a fact that is generally taught in class and not expected to be derived (based on my limited experience searching for similar courses). Were you not taught that the arrival times are i.i.d. uniform on $$[0,T]$$ conditioned on the number of arrivals up to a time $$T$$? It is 2.4.3 on Durrett EOSP (available online free). The proof of it is not bad; you just write out the conditional, and note that it does not depend on the exact values of the arrival times (Normally, the joint density of the $$k$$ arrival times only depends on the last one, and here, all we know is that the $$k+1$$th arrival is larger than $$T$$). With this knowledge at hand, I am sure you can do that first part.

Your second one is exactly right.

EDIT: Forgot the word i.i.d., so added that, and adding a couple guiding questions for the first part:

a) Say the names of the planes that landed are A, B, C (not necessarily in landing order, so they could have landed A - C - B, say) What is the probability that $$B$$ landed in the first hour? b) What is the probability that both $$B$$ and $$C$$ landed in the first hour?

• We have had the proof, that is right, I am not sure how to use this fact though?
– user459879
Commented Jun 2, 2021 at 17:54
• OK, so 1/2 of the time is right. So now, you have 3 independent planes, each having 1/2 probability of landing in the first hours, and 1/2 in second. What is the probability that they all land in second hour?
– E-A
Commented Jun 2, 2021 at 18:08
• They all land in the second hour with probability $\left(\frac{1}{2} \right)^3 = \frac{1}{8}$. This is the complement of the event that at least 1 lands in the first hour, so then that would be $\frac{7}{8}$? Which is the probability I think I am interested in.
– user459879
Commented Jun 2, 2021 at 18:11
• yup good job! good work.
– E-A
Commented Jun 2, 2021 at 18:12
• Thank you, you did a lot of steering though, Really. You've been most kind.
– user459879
Commented Jun 2, 2021 at 18:13