You and N of your friends are meeting for dinner, where N is a Poisson random variable with . All of you arrive independently and according to a uniform distribution over (0,1). Find the expected number of friends that arrive before you.
my book has an example, but they skip a lot of steps, so all I have is:
and they conclude that it's but I'm a little lost at picking my E[N](t)
The hypothesis about the number and the arrival times of the friends is equivalent (this is a theorem) to saying that the friends arrive according to a Poisson point process of intensity , stopped at time 1.
Then, the number of friends that arrive before time equals the number of points of the point process before . As a consequence, is a Poisson random variable of parameter , and .
Finally, let us denote by your arrival time (uniformly distributed on [tex][0,1][/Math]). Because of the independence between you and your friends, the expected number of friends that arrive before you is .