Random Walk - Expected Time Until Absorption

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Homework Statement



Consider the following random walk on the integers:

[tex]\mathbb{S}=\{0, 1, 2, 3, ... , L\}[/tex]

Let Wn = {the state k [tex]\in\mathbb{S}[/tex] you are in after the n'th transition}

[tex]\mathbb{P}[W_{n+1}= k \pm 1 | W_{n} = k] = \frac{1}_{2}[/tex]

[tex]For \ 1\leq k\leq L-1[/tex]

Otherwise:

[tex]\mathbb{P}[W_{n+1}= L | W_{n} = L] = \frac{1}{2} = \mathbb{P}[W_{n+1}= L -1 | W_{n} = L] [/tex]

[tex]\mathbb{P}[W_{n+1}= 0 | W_{n} = 0] = 1[/tex]

i.e. L is retaining, and 0 is absorbing.

Determine:

[tex]\mathbb{E}[T | W_{0}=k] \forall \ k \in \mathbb{S}[/tex]

[tex]Where \ T=min \{n\geq 0 : W_{n}=0 \}[/tex]

Not sure how to proceed...
 

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