Consider a random walk on [0, inf) where you start at 0. With probability p = 0.5, you increase by 1. With probability (1-p) = 0.5, you decrease by 1, but not below 0.

As time goes to infinity, will your position tend to infinity? If not, to what finite value does it converge?

Edit: To be a bit more precise, what is the limit of the average position as time goes to infinity?

If your position tends to infinity for p = 0.5, for which other probabilities p is this true? (Clearly p > 0.5 will cause you to tend to infinity, so p < 0.5 is what I'm after)

What is the probability of being at position x after an arbitrary amount of steps?

I made a simple simulation to test the p = 0.5 case, and after 2 500 million iterations, it seems to tend to infinity, but I'd like a more solid explanation.

Thanks!

Consider a random walk on [0, inf) where you start at 0. With probability p = 0.5, you increase by 1. With probability (1-p) = 0.5, you decrease by 1, but not below 0.

As time goes to infinity, will your position tend to infinity? If not, to what finite value does it converge?

Edit: To be a bit more precise, what is the limit of the average position as time goes to infinity?

If your position tends to infinity for p = 0.5, for which other probabilities p is this true? (Clearly p > 0.5 will cause you to tend to infinity, so p < 0.5 is what I'm after)

What is the probability of being at position x after an arbitrary amount of steps?

I made a simple simulation to test the p = 0.5 case, and after 2 million iterations, it seems to tend to infinity, but I'd like a more solid explanation.

Thanks!

Consider a random walk on [0, inf) where you start at 0. With probability p = 0.5, you increase by 1. With probability (1-p) = 0.5, you decrease by 1, but not below 0.

As time goes to infinity, will your position tend to infinity? If not, to what finite value does it converge?

If your position tends to infinity for p = 0.5, for which other probabilities p is this true? (Clearly p > 0.5 will cause you to tend to infinity, so p < 0.5 is what I'm after)

What is the probability of being at position x after an arbitrary amount of steps?

I made a simple simulation to test the p = 0.5 case, and after 2 million iterations, it seems to tend to infinity, but I'd like a more solid explanation.

Thanks!