Originally Posted by

**usagi_killer** Hi romsek,

I just want to get this question clearer, as I'm not sure how to exactly interpret the question.

So, define $U_k$ as the random variable that denotes the amount of petrol that car $k$ fills, $k = 1, 2, 3, \cdots$. Thus, $U_k, k = 1, 2, 3, \cdots$ are independently and identically distributed as a uniform random variable over $(0,50)$.

Let $N$ be the random variable that denotes the number of cars that the petrol station can *fully* service.

Now I don't quite get the question. Say we have the following scenario:

Car 1 comes with 10L remaining in its tank, so it will fill up 40L, hence the amount of petrol left in the station is now 100-40 = 60L.

Car 2 comes with 10L remaining in its tank, so it will fill up 40L, hence the amount of petrol left in the station is now 60-40 = 20L.

Car 3 comes with 20L remaining in its tank, so it will fill up 30L, but the petrol station only has 20L left, so does this mean Car 3 just leaves the petrol station filling 0L? My gut feeling is that this cannot happen because each car can only fill an amount BETWEEN 0 and 50, ie, (0,50) [note that the end points are not included].

Hence in this scenario, the petrol station runs "out" of petrol at N=3 because it does not have enough to FULLY service Car 3, even though it still has 20L left in the pump. Thus, the petrol station can only service N=2 cars.

Is this interpretation correct?

Also how did you compute the infinite sum for the expectation? And can you give me a guide on how I can derive the distribution of $G_n$?