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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:wt(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|w1=1+1]P[w1=1+1] + E[S|w1=1-1]P[w1=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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