- #1

fignewtons

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## Homework Statement

Let w(1) = event of a random walk with right drift (p > q, p+q = 1) starting at 1 returns to 0

Let p(w(1)) = probability of w(1)

Let S=min{t>=0:w

_{t}(1)=0} be the minimum number of steps t a walk starting from 1 hits 0.

What is E[S|w(1)]?

## Homework Equations

I know E[S|w(0)] = 0 (expected # of steps to get from 0 to 0 is 0)

## The Attempt at a Solution

E[S|w(1)]=E[S|w

_{1}=1+1]P[w

_{1}=1+1] + E[S|w

_{1}=1-1]P[w

_{1}=1-1]

=[1+E[S|w(2)]]p+[1+E[S|w(0)]]

=pE[S|w(2)] + qE[S|w(0)] + 1

=pE[S|w(2)] + 1

I'm not sure how to proceed. Any tips? Please be specific and try to be as simple as possible. I'm new to this.

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