Let p>q>0 with p+q = 1 and a = q/p < 1.
Let X_n denote the random walk with transitions
X_{n+1} = CASE 1: X_n + 1 with probability p and CASE 2: X_n - 1 with probability q.
For i ≥ 0, we set u_i = P(X_n = 0 for some n ≥ 0|X_0 = i).
Give the value of u_0.
To find the value of u_0, we need to determine the probability that the random walk X_n, starting from X_0 = 0, reaches 0 at some point.
Let's analyze the possible transitions for the random walk X_n.
If X_0 = 0, there are two possible cases:
1. X_1 = 1 with probability p: In this case, we can't reach 0 in the next step.
2. X_1 = -1 with probability q: In this case, we can reach 0 in the next step.
Since p > q and p + q = 1, the probability of transitioning to -1 (Case 2) is greater than the probability of transitioning to 1 (Case 1). This implies that the random walk X_n is more likely to go downwards (towards -1) rather than upwards (towards 1).
Now, let's consider the probability of reaching 0. Since reaching 0 is dependent on the previous step, we can calculate the probability recursively.
Starting with X_0 = 0, the probability of reaching 0 at some point can be expressed as follows:
u_0 = probability of reaching 0 in the next step (Case 2: X_1 = -1) + probability of reaching 0 after multiple steps from -1.
Since the random walk X_n is more likely to go downwards, we can assume that the probability of reaching 0 after multiple steps from -1 is equal to u_0.
Therefore, we have the equation:
u_0 = q + u_0
To solve for u_0, we can rearrange the equation:
u_0 - u_0 = q
0 = q
Hence, the value of u_0 is q.
In summary, the value of u_0 is q, which is equal to the probability of transitioning to -1 from X_0 = 0.