I need help to solve these exercises about Poisson processes:
1)Arrivals of the Number 1 bus form a Poisson process of rate one bus per hour, and arrivals of the Number 7 bus form an independent Poisson process of rate seven buses per hour.
a) What is the probability that exactly three buses pass by in one hour
b) What is the probability that exactly three Number 7 buses pass by while I am waiting for a Number 1?
c) When the maintenance depot goes on strike half the buses break down before they reach my stop. What, then, is the probability that I wait for 30 minutes without seeing a single bus?
2)A radioactive source emits particles in a Poisson process of rate λ. The particles are each emitted in an independent random direction. A Geiger counter placed near the source records a fraction p of the particles emitted. What is the distribution of the number of particles recorded in time t?
3)A pedestrian wishes to cross a single lane of fast-moving traffic. Suppose the number of vehicles that have passed by time t is a Poisson process of rate lambda and suppose it takes time a to walk across the lane. Assume that the pedestrian can foresee correctly the times at which vehicles will pass by and that the pedestrian will not cross the lane if, at any time whilst crossing, a car would pass in either direction.
a) How long on average does it take to cross over safely?
How long on average does it take to cross two similar lanes
b) when one must walk straight across?
c) when an island in the middle of the road makes it safe to stop halfway?