Problem 724

[KING-OLE]

Not sure I understand the question. Are you looking for the distance of:

1. All packages added
2. The package that has traveled the longest
3. The average distance all packages has traveled

Thanks.

[mdean]

If this helps, the expected distance means the expected value of the distance. I believe it is also assumed that each drone is equally likely to be chosen each second.

[heteroing]

The problem wants the expectation of the average distance of all drones once the process completes.

[KING-OLE]

The problem wants the expectation of the average distance of all drones once the process completes.

Thanks, and I must have been tired last night, as the only way to get to a non-integer is by taking the average.

[jarip]

I must be miss-understanding something , and failing the the first test case.
Let's call the two drones (H,T):
Is the following correct for the process:
- at time t=0, a fair coin is flipped to select a drone, let's assume it was H
- at time t=1, another flip is made:
- if T , drone T starts and process completes at t=2,
total distance = 3, drone H has traveled 2 cm and drone T, 1cm
- flip at time t=2
- if H , drone H increases its speed to 2 cm/sec
- if T, the process terminates at t=3 with total distance (1 + 2 + 2 ) + (1) = 6,
otherwise drone H increases speed to 3,
at next possible termination, t=4 the total distance is ( 1 + 2 + 3 + 3) + 1 = 10
In the other scenario T starts first.

- If the above was correct, the expectation for the distance is then calculated as:
P(completes t=2)*3 + P(completes t=3)*6 + P(t=4)*10 + ...
Here P(completes t=2) = P(HT) + P(TH) = 2/4, and P(t=3) P(HHT) + P(TTH) = 2/8, etc.

[jarip]

Actually I think I see my problem,
I am calculating the expectation for the sum of the traveled distance,
but it seems what is asked is the expectation for the average distance.

[LangerJan]

Are jarpis assumptions about how the drones behave correct?

Given that there are two drones (#0 and #1):

If I expect that drone #1 is contacted at the third try, the following happens:

Time: 0s:

• #0, speed 0cm/s, traveled 0cm
• #1, speed 0cm/s, traveled 0cm

Depot calls #0, its speed increases to 1cm/s

Time: 1s:

• #0, speed 1cm/s, traveled 1cm
• #1, speed 0cm/s, traveled 0cm

Depot calls #0, its speed increases to 2cm/s

Time: 2s:

• #0, speed 2cm/s, traveled 3cm
• #1, speed 0cm/s, traveled 0cm

Depot calls #1, its speed increases to 1cm/s

Time: 3s:

• #0, speed 2cm/s, traveled 5cm
• #1, speed 1cm/s, traveled 1cm

Both drones drop their package. They traveled 6cm in total, on average 3cm.