Timeline for Hitting time probability in a Random Walk with possibility to die.
Current License: CC BY-SA 3.0
15 events
| when toggle format | what | by | license | comment | |
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| Dec 12, 2013 at 14:56 | comment | added | Hans | I have now edited the original question to add "and die" after "jump again to the starting point ...". Otherwise the statement is vague. | |
| Dec 12, 2013 at 14:38 | comment | added | Hans | By the way, the misinterpreted version where the particle does not die but restarts from the starting point is in itself an interesting problem in computing the expected hitting time of the boundary. | |
| Dec 12, 2013 at 14:36 | comment | added | Hans | My apology. I did not see the word "die" in the title until now that you referred me to it. As I read the original question in the content, I interpreted it as the particle jump back the starting point to restart the journey instead of to "die". I think the original poser should have repeat the word in the full statement of the problem. The reason I use the the word "impression" was that I wanted to delay my judgment, even though I thought --- mistakenly --- you were misreading the question, and to give us the chance to fully discuss the meaning of the problem. My apology again! | |
| Dec 12, 2013 at 7:58 | comment | added | Douglas Zare | @Hansen: You don't see "die" in the title of the question, or you don't think the title means anything? You can't figure out what I meant by die, and you need me to spell out the only thing which makes sense again? Meanwhile, you repeatedly say that I'm wrong without actually finding any error. If you actually find a mistake, let me know, but I'm not interested in your vague impressions or confusion. Good bye. | |
| Dec 12, 2013 at 5:25 | comment | added | Hans | To put it more succinctly, where is death in the original question? | |
| Dec 12, 2013 at 5:25 | comment | added | Hans | My current impression is that that is wrong, but I will reserve my judgment until you clarify your derivation. I think you have answered a different question than the OP asks. The particle never dies but only restarts from the starting point in the original question whereas you change the statement to have the particle die. I think this is wrong unless you can explain the death version is equivalent to the original version, which I think is an impossible task. | |
| Dec 12, 2013 at 3:02 | comment | added | Douglas Zare | @Hansen: I think my answer is clear. I don't know why you misread it. I computed $s$ and $d$ and stated that the expected number of steps before hitting a barrier is $\frac{s}{1-d}$. Are you still saying that's wrong? | |
| Dec 11, 2013 at 17:40 | comment | added | Hans | So the expected time asked in the question is longer than what you have computed. | |
| Dec 11, 2013 at 16:59 | comment | added | Hans | You are right that the probability of reaching the boundary in the original setting is 1. The original question asks for not the probability but the expected time of reaching the boundary. The right recursion equation to look at therefore is what you call b[k], the expected number of step reaching the boundary but excluding the starting point. The particle does not die. The correct equation of b is: b[k] = 1+ p b[k+1]+q b[k-1]+r b[s], where s is the starting position. This can be solved with generating function. | |
| Dec 11, 2013 at 16:06 | comment | added | Hans | If that is how you meant "death" to be, you should have stated it explicitly in your answer. Are you letting the particle disappear or die once it jumps back to the starting point? That is not what the original question asks. The original question stipulates that the point restarts once it comes back to the starting point, and it asks for the expected time to reach the boundary (one of the two barriers) rather than the question you modifies to which is the expected time to reach the boundary OR the starting point. So your answer does not answer the original question. | |
| Dec 11, 2013 at 7:05 | comment | added | Douglas Zare | @Hansen: Death means jumping back to the start. I computed the probability that you make it to the boundary without dying, and the expected number of steps before you die or restart. If you calculate the recurrence you suggest, that's the probability of making it to the boundary at some point, which is $1$ when the system is not degenerate, hence not an interesting thing to compute. That $1$ is not what you need to compute the expected exit time from the expected value of the steps to reach the boundary or restart. | |
| Dec 11, 2013 at 2:50 | comment | added | Hans | The correct recursion for a is: a[k] = p a[k+1]+q a[k-1]+r a[0]. One can solve it with a generating function. | |
| Dec 11, 2013 at 2:18 | comment | added | Hans | You have never defined death. The OP certainly has not defined it. The recursion equation of a[k] seems to suggest that you consider jumping back to k=0 makes a[0], presumably the probability of reaching either -1 or 1, zero. That is wrong. | |
| Mar 28, 2013 at 8:16 | comment | added | Douglas Zare | See also mathematica-journal.com/2012/03/… | |
| Mar 28, 2013 at 8:07 | history | answered | Douglas Zare | CC BY-SA 3.0 |