how long on average would it take for a person to randomly step on any point $n$ times. how long on average would it take for a person to randomly step on any point $n$ times.
This person walks in a random way every odd step: he can choose to go north or south $1$ meter.
Every even step, he walks 1 meter east or west.
My question is, on average, how many steps will it take to go to any point $n $ times?
I would like to learn how to solve problems like this. I'm a 9th grader hoping to learn probability.
Your walk is essentially two independent random walks, so let's just consider one of them, i.e. a (symmetric) random walk on the integers.
You are asking about the expected time of the $n$th visit to any arbitrary point. Let's first try to answer the problem, but with the point being the starting point.
It turns out that you can show that such a walk is recurrent (the probability of return is 1), but not positive-recurrent (the expected (1st) return time is infinite).
You can learn more by consulting these lecture notes which i found nice http://www.statslab.cam.ac.uk/~rrw1/markov/M.pdf
Your walk is essentially two independent random walks, so let's just consider one of them, i.e. a (symmetric) random walk on the integers.
You are asking about the expected time of the $n$th visit to any arbitrary point. Let's first try to answer the problem, but with the point being the starting point.
It turns out that you can show that such a walk is recurrent (the probability of return is 1), but not positive-recurrent (the expected (1st) return time is infinite).
You can learn more by consulting these lecture notes which i found nice http://www.statslab.cam.ac.uk/~rrw1/markov/M.pdf