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I have a symmetric random walk on the integers with probability $p$ and $q$ of going up and down respectively started at $X_0 = 2$.

Let

$$ T^0 = \{ n > 0: X_n = 0\}, T^1 = \{ n > 0: X_n = 1\} $$$$ T^0 = \min\{ n > 0: X_n = 0\}, T^1 = \min\{ n > 0: X_n = 1\} $$

I want to show that the expected time from 2 to hit zero is the expected time of hitting one plus the expected time of hitting 0 from 1. $$ \mathbb{E}_2[T^0] = \mathbb{E}_2[T^1] + \mathbb{E}_1[T^0] $$

This seems intuitively obvious to me, but I would like to be able to show it formally. I think this is a job for the law of iterated expectations, but the conditioning on the starting position confuses me. Can anyone show how to prove this formally step by step?

I have a symmetric random walk on the integers with probability $p$ and $q$ of going up and down respectively started at $X_0 = 2$.

Let

$$ T^0 = \{ n > 0: X_n = 0\}, T^1 = \{ n > 0: X_n = 1\} $$

I want to show that the expected time from 2 to hit zero is the expected time of hitting one plus the expected time of hitting 0 from 1. $$ \mathbb{E}_2[T^0] = \mathbb{E}_2[T^1] + \mathbb{E}_1[T^0] $$

This seems intuitively obvious to me, but I would like to be able to show it formally. I think this is a job for the law of iterated expectations, but the conditioning on the starting position confuses me. Can anyone show how to prove this formally step by step?

I have a symmetric random walk on the integers with probability $p$ and $q$ of going up and down respectively started at $X_0 = 2$.

Let

$$ T^0 = \min\{ n > 0: X_n = 0\}, T^1 = \min\{ n > 0: X_n = 1\} $$

I want to show that the expected time from 2 to hit zero is the expected time of hitting one plus the expected time of hitting 0 from 1. $$ \mathbb{E}_2[T^0] = \mathbb{E}_2[T^1] + \mathbb{E}_1[T^0] $$

This seems intuitively obvious to me, but I would like to be able to show it formally. I think this is a job for the law of iterated expectations, but the conditioning on the starting position confuses me. Can anyone show how to prove this formally step by step?

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Three Diag
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  • 8

Explain formally how expected time to hit 0 from two is the sum of the expected time to hit 1 from 2 and 0 from 1

I have a symmetric random walk on the integers with probability $p$ and $q$ of going up and down respectively started at $X_0 = 2$.

Let

$$ T^0 = \{ n > 0: X_n = 0\}, T^1 = \{ n > 0: X_n = 1\} $$

I want to show that the expected time from 2 to hit zero is the expected time of hitting one plus the expected time of hitting 0 from 1. $$ \mathbb{E}_2[T^0] = \mathbb{E}_2[T^1] + \mathbb{E}_1[T^0] $$

This seems intuitively obvious to me, but I would like to be able to show it formally. I think this is a job for the law of iterated expectations, but the conditioning on the starting position confuses me. Can anyone show how to prove this formally step by step?