Consider a random walk on the integers in [0, n] where on...

Consider a random walk on the integers in [0, n] where on each move, one moves right with probability p ? 1/2 and left with probability q = 1 - p. The endpoints 0 and n are absorbing. Show that the probability sk that a walk starting at x0 = k is absorbed at 0 is sk = ((q/p)^k - (q/p)^n) / (1 - (q/p)^n)

Transcript

00:01 So to begin, we can note that the probability of being absorbed at a particular point, or pardon me, the probability of being absorbed at zero, given that we start at x not equals k, will be equal to the probability that we move to the left, so that's q or 1 minus p, times the probability that we get absorbed at zero, given that we start at k minus 1, plus p times the probability that we get absorbed at zero, given that we start at k plus one.

00:39 So we can now turn this into a standard difference expression, or standard difference equation form, writing this as sk plus 1 is equal to 1 minus p times sk minus 1, minus sk all over p.

01:01 So that's 1 minus p, or sk minus 1, minus sk over p.

01:06 Or alternatively, that's q over p times sk minus sk minus sk over p.

01:18 Now i'll introduce the sort of guess form that sk is going to be equal to a times r to the power of k.

01:29 In that case, we would have that a times r to the power of k plus 1 is equal to q over p times a r to the power of k minus 1 minus 1 over p times a r to the power of k times a r to the power of k.

01:50 Dividing everything by a r to the power of k will give us r equals q over p times r to the of negative 1 minus 1 over p or actually pardon me i just realized it's going to make more sense to divide everything by k or a r to the power of k minus 1 rather so this will give us r squared equals q over p times r minus 1 over p times or slight mistake that should be q over p just times 1 technically then we have minus 1 over p times r so we now have that this is going to be solved in the form of quadratic.

02:34 So we'll have that this has two solutions.

02:36 R equals negative 1 plus or minus the square root of 1 plus 4 pq divided by, or all over, 2 times p.

02:50 So knowing the different possible r values here, for simplicity, i'll just refer to these as r plus and r minus referring to if we're taking the plus or the minus option.

03:03 But we can now see that we have that sk is going to be in the form of a times, or, pardon me, it would be in the form of c1, r plus to the power of k, plus c2, r minus to the power of k.

03:21 Now we need to impose our initial conditions in order to find our coefficients.

03:26 So we'd have that, you know, if we start at zero, then the probability of being absorbed into 0 is going to be equal to 1...

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