Modified Random Walk: Expected Duration and Recurrence Equation Analysis

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Poirot1
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(1)Consider the expected duration of the modified random walk. Show that conditioning on the first step produces a recurrence equation of the following form.​
Ea = ˜pEa+1 + ˜qEa−1 + ˜c.
[FONT=CMR12](2)Clearly identify the values of ˜[FONT=CMMI12]p[FONT=CMR12], ˜[FONT=CMMI12]q [FONT=CMR12]and ˜[FONT=CMMI12]c [FONT=CMR12]in terms of [FONT=CMMI12]p[FONT=CMR12], [FONT=CMMI12]q [FONT=CMR12]and [FONT=CMMI12]r[FONT=CMR12].
Thanks
 
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Poirot said:

(1)Consider the expected duration of the modified random walk. Show that conditioning on the first step produces a recurrence equation of the following form.​
Ea = ˜pEa+1 + ˜qEa−1 + ˜[FONT=CMMI12]c.
[FONT=CMR12](2)Clearly identify the values of ˜[FONT=CMMI12]p[FONT=CMR12], ˜[FONT=CMMI12]q [FONT=CMR12]and ˜[FONT=CMMI12]c [FONT=CMR12]in terms of [FONT=CMMI12]p[FONT=CMR12], [FONT=CMMI12]q [FONT=CMR12]and [FONT=CMMI12]r[FONT=CMR12].
Thanks


Please post the entire question, in particular the definition of your modified random walk

CB