What is the expected delay between the grey buses?

Question

Four yellow buses and two grey buses that could be in any order (with equal probability) are traveling together with the probability of a delay of at least $\mathit{t}$ seconds between any two consecutive buses being $\mathit{P((t,\infty)) = e^{-t} }$ $\forall$ $\mathit{t \ge 0}$ , independent of all other delays. What is the expected delay between grey buses?

I know that this problem deals with total expectation, so I will have to have several random variables, such as G for the grey buses and Y for the yellow, and then use the fact that I know the probability of the delay between any two buses to find the delay between the two grey buses.

Really, I'm not sure how to set this up so that it works. Is the distribution between any two buses an exponential RV? If that were the case, could we easily find conditional distributions and expectations to work with?

Any help is appreciated!

Answer 1

Hints:

• What is the expected length of a gap between consecutive buses? This is indeed an exponential random variable

• What is the expected number of gaps between grey buses? You might find this easier if you consider the expected number of yellow buses between the two grey buses and perhaps the probability a given yellow bus comes between the grey buses, or you could just look at all the possible patterns

Answer 2

Hint:

1. There can be $0,1,2,3,4$ different yellow buses between two different grey buses, try to find probability of each case.

2. Identify the distribution, then you can easily find expectation between two consecutive buses. All the delays are independent, so you can find answer.