Consider a random walk on 0, ..., k, which moves left and...

Consider a random walk on {0, ..., k}, which moves left and right with respective probabilities q and p. If the walk is at 0 it transitions to 1 on the next step. If the walk is at k it transitions to k - 1 on the next step. This is called random walk with reflecting boundaries. Assume that k = 3, q = 1/4, p = 3/4, and the initial distribution is uniform. For the following, use technology if needed. (a) Exhibit the transition matrix. (b) Find P(X7 = 1|X0 = 3, X2 = 2, X4 = 2). (c) Find P(X3 = 1, X5 = 3).

Transcript

00:01 Okay, so we've got a random walk here.

00:03 In the case that we're being told, we've got three steps in it.

00:06 So we've got position zero, position one, position two and position three.

00:10 And it says that if you're at position zero, you have to go to position one.

00:18 And if you're at position three, you have to go to position two.

00:21 Otherwise, the probability to move left is one quarter, and the probability to move right is three quarters.

00:27 You don't stay still.

00:29 So part a asks us to find the transition matrix.

00:32 So this is basically our four possible states.

00:35 We could be in position 0, 1, 2 or 3.

00:38 And so our transition matrix is going to be the probability to transition from the start state, which is the row label, to the final state, which is the column label.

00:57 So for instance, this entry here is the probability to start at in state zero and end in state zero.

01:04 But this is zero because you can't see.

01:06 Stay still.

01:07 We know that if you're at position zero, you have to move to position one.

01:09 So that happens with probability one and the others happen with probability zero.

01:13 If you're at position one, you could move left to position zero with probability one quarter.

01:19 You could move right, which will get you up to position two with probability three quarters.

01:23 And then you can't stay the same or get to position three because that's two jumps away.

01:27 If you start at state two, you could move left to state one with a quarter with probability a quarter, right to state three with probability three quarters and couldn't get to be the other two.

01:35 And then if you're at state three, you have to move to state two.

01:38 So that happens with probability one and the others are all zero.

01:41 So this is our transition matrix.

01:45 The next question is a bit of a silly one because it says to, it wants you to find the probability that x7 is one given that x0 is 3, x2 is 2 and x4 is 1.

02:02 I think what they're saying is here that after i steps, you're at position xi...

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