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Probability generating functions

This isn't really the original Achilles & tortoise problem but it uses the characters:

Suppose that the race began with Achilles waiting on the starting line while the tortoise was given a start of t time units, where t is a positive integer. Suppose also that, during each time unit, the tortoise either moves 1 metre with probability p or stays where it is with probability 1-p. Write down the distribution of the distance travelled by the tortoise before it fails to move and find the probability generation function.

I could do this part because it's the geometric distribution, and the pgf is (1-p)/(1-sp).

Once he has started, Achilles runs at a constant rate of 1m per time unit. Find the pgf for the distance from the start at which Achilles catches the tortoise and show that its expectation is tp/(1-p).

I'm really stuck on this part.. Suppose the tortoise travels D metres before Achilles starts. Achilles will need to travel each time the tortoise travels, plus all the times the tortoise fails to travel, and there need to be D of these, so we need to find the amount of time it takes the tortoise to fail D times? It looks like a sum of I.I.D. random variables but I'm not sure how to go about it. Please help?
Reply 1
Avatar for DFranklin
DFranklin
18
Basically, every time the tortoise fails to move, Achilles catches up one unit. And you know he needs to catch up t units. So it is exactly a sum of IID random variables (of the form you were asked about in the first part of the question).

What exactly, are you stuck on?
Reply 2
Avatar for jessica15
jessica15
OP
Original post by DFranklin
Basically, every time the tortoise fails to move, Achilles catches up one unit. And you know he needs to catch up t units. So it is exactly a sum of IID random variables (of the form you were asked about in the first part of the question).

What exactly, are you stuck on?


Why does he need to catch up t units? The tortoise has not necessarily moved t units in the headstart, so the number of units he needs to catch up on looks like a random variable as well?
Reply 3
Avatar for DFranklin
DFranklin
18
I haven't done the calcs, so this is possibly wrong, but think in terms of catching up *time* units, rather than distance.

[That is, each time the tortoise fails to move, he's "wasting" a unit of time, relative to Achilles].
Reply 4
Avatar for jessica15
jessica15
OP
Original post by DFranklin
I haven't done the calcs, so this is possibly wrong, but think in terms of catching up *time* units, rather than distance.

[That is, each time the tortoise fails to move, he's "wasting" a unit of time, relative to Achilles].


But each time unit corresponds to Achilles catching up 1 distance unit. He has a headstart of D~Bi(t, p) units, of Achilles needs to catch up on D time (or, equivalently, distance) units? I don't think he needs to catch up t units because the tortoise is not necessarily that far ahead. Maybe there's something I'm missing? :s-smilie:

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