Help needed on a dp problem about probability and expectation

Revision en2, by Gnay_Oahnauhz, 2016-05-05 10:51:02

The problem is like this: You are on the origin of a number axis, each time, you can move forward, backward or stay with probability Pf, Pb, Ps = 1 - Pf + Ps respectively, i.e., if you are on x now, the next step you can move to x + 1 with probability of Pf and etc. And the question is: after n steps, what's the mathematical expectation of the maximal number you have reached.

Here's an example when n = 2, Pf = 0.25, Pb = 0.25, Ps = 0.5:

maximal number: 1
fs: 0.25*0.5 = 0.125
sf: 0.5*0.25 = 0.125
fb: 0.25*0.25 = 0.0625 (since you have arrived at number 1, even if you go back, you will have maximal number 1)
maximal number: 2
ff: 0.25*0.25 = 0.0625

since you are on the origin, all other path will have maximal number equal to 0

Thus the expectation is: 1*(0.125+0.125+0.0625)+2*0.0625=0.4375

There is a dp solution on the problem, but I just can't quite figure out why, perhaps you can just think it over first before I give the answer.

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en4 English Gnay_Oahnauhz 2016-05-05 19:03:52 244 (published)
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