Complete question is;
Fred the ant is on the real number line, and Fred is trying to get to the point 0
If Fred is at 1 then on the next step, Fred moves to either 0 or 2 with equal probability. If Fred is at 2 then on the next step, Fred always moves to 1
Let e1 be expected number of steps Fred takes to get to 0 given that Fred starts at the point 1. Similarly, let e2 be expected number of steps Fred takes to get to 0 given that Fred starts at the point 2
Determine the ordered pair (e1, e2)
Answer:
(e_1,e_2) = (2,3)
Step-by-step explanation:
We can track the probabilities using the scenario that Ant gets to 0 after 1 steps, 2 steps, 3 steps, 4 steps, 5 steps and so on.
This gives us e_1 = 1(1/2^(1)) + 2(1/2²) + 3(1/2³) + 4(1/2⁴) + ...
By arithmetico-geometric series, which is given by;
S_(∞) = [a/(1 - r)] + dr/(1 - r)²
From the online calculator, i got;
e_1 = 2
Similarly, e_2 = 2(1/2^(1)) + 3(1/2²) + 4(1/2³) + 5(1/2⁴) + ...
By arithmetico-geometric series, e_2 = 3,
Thus, (e_1,e_2) = (2,3).
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