Fred the ant is on the real number line, and Fred is trying to get to the point 0.
If Fred is at 1 then on the next step, Fred moves to either 0 or 2 with equal probability. If Fred is at 2 then on the next step, Fred always moves to 1.
Let \(e_1\) be expected number of steps Fred takes to get to 0 given that Fred starts at the point 1. Similarly, let \(e_2\) be expected number of steps Fred takes to get to 0 given that Fred starts at the point 2.
Determine the ordered pair \((e_1, e_2)\).
Thanks in advance for anyone who can help! Thank you!
You can track the probabilities using different case, i.e. Anty gets to 0 after 1 steps, 2 steps, 3 steps, and so on. This gives us e = 1/2*1 + 1/4*2 + 1/8*3 + 1/16*4 + ... By arithmetico-geometric series, e = 2. Similarly, f = 1/2*2 + 1/4*3 + 1/8*4 + 1/16*5 + ... = 3, so (e,f) = (2,3).
thank you guest for your effort and time, but I think that (2,3) may be wrong. I don't know what is wrong with it, since I think that the reasoning is quite clear, but I will take a few moments to see what could be wrong with it.
I'm curious too.
Alan's post is being blocked here but available on the answer page for some weird reason
so I have taken a pit of it and will try to display it here.
Alan, can you explain more how you got from second to last step to the answer? I don't seem to quite understand. :( I do understand the beginnning and middle though. Good reasoning!!!
Thanks Alan ... but I do not know how to calculate that sum to get 3 either.
If anyone can answer please, what is the concept of \(\lambda \) in sums that Alan used called?
Thanks very much Alan,
I am never quite confident about expected values. :)