# Probability generating functions - please spot my mistake!

• Jan 30th 2009, 06:37 AM
James0502
Probability generating functions - please spot my mistake!
According to Zeno (495-435 BC) there was a race in which Achilles had to catch a tortoise having given it a start. Suppose that the race began with Achilles waiting on the starting line while the tortoise was given a start of r time units, where r is a positive integer. Suppose also
that, during each time unit, the tortoise either moves 1 metre with probability p and stays where
it is with probability q (p + q) = 1. Write down the distribution of the distance travelled by the tortoise
before it rst fails to move and fi nd the probability generating function.

Generating function: qs/(1-ps)
Once he has started, Achilles runs at a constant rate of 1 metre per time unit.

Find the probability generating function for the distance from the start at which Achilles catches the tortoise and show that its expectation is rp/q

My answer: Find where tortoise has not moved r times

Binomial distribution, parameter q

generating function: (p + qs)^n

I have put my first genearting function is this one, and differentuiated, and put s = 1 to give just n, so I mustve made a mistake

Many Many Thanks - I've been stuck on this for days!
• Jan 30th 2009, 03:30 PM
awkward
Quote:

Originally Posted by James0502
According to Zeno (495-435 BC) there was a race in which Achilles had to catch a tortoise having given it a start. Suppose that the race began with Achilles waiting on the starting line while the tortoise was given a start of r time units, where r is a positive integer. Suppose also
that, during each time unit, the tortoise either moves 1 metre with probability p and stays where
it is with probability q (p + q) = 1. Write down the distribution of the distance travelled by the tortoise
before it rst fails to move and fi nd the probability generating function.

Generating function: qs/(1-ps)
Once he has started, Achilles runs at a constant rate of 1 metre per time unit.

Find the probability generating function for the distance from the start at which Achilles catches the tortoise and show that its expectation is rp/q

My answer: Find where tortoise has not moved r times

Binomial distribution, parameter q

generating function: (p + qs)^n

I have put my first genearting function is this one, and differentuiated, and put s = 1 to give just n, so I mustve made a mistake

Many Many Thanks - I've been stuck on this for days!

I think you may need the negative binomial distribution on the second part, not the binomial distribution.