The number of hours between successive train arrivals at the

The number of hours between successive train arrivals at the station is uniformly distributed
on [0,1]. Passengers arrive according to a Poisson process N t with rate 7 per hour. Suppose a
train has just left the station. Let X denote the number of people who get on the next train. Find
(a) E[X]
(b) Var[X]

Solution

Consider the next train will arrive at time t > 0 then we can say that the number of customers at time t is a Poisson random variable with parameter t.

Therefore, using T as the random variable of the next train arrival:

E[X] = E[ E [ X | T ] ] = E[ T ] = 7 / 2 = 3.5.

Var [X] = Var [ Var [ X | T] ] = ² / 12 + / 2 = 49 / 12 + 7 / 2 = 7.583333