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Drunkard's walk's solution: significance of continuity
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I have trouble understanding solution of the Drunkard's walk problem. Here is the original statement of the problem:
There once was a drunk man who wandered far too close to a cliff. From where he stands, one step forward would send the drunk man over the edge. He takes random steps, either towards or away from the cliff. At any step, his probability of taking a step away is 2/3 and a step towards the cliff is 1/3.
What is his chance of escaping the cliff?
From the book that I read, a solution involves proving continuity of some function. Here's a relevant portion of the solution:
What I don't understand is, why continuity comes into play here? Solution from other sources (example) that also involve solving for P1 don't mention continuity at all. It seems to me that the author really stresses the importance of continuity, whose proof he skips by the way ("beyond the scope of this book").
probability continuity random-walk
edited Jul 24 at 15:38
Mike Pierce
11k93574
asked Jul 23 at 4:38
Chav Likit
495
add a comment |Â
up vote
1
down vote
favorite
I have trouble understanding solution of the Drunkard's walk problem. Here is the original statement of the problem:
There once was a drunk man who wandered far too close to a cliff. From where he stands, one step forward would send the drunk man over the edge. He takes random steps, either towards or away from the cliff. At any step, his probability of taking a step away is 2/3 and a step towards the cliff is 1/3.
What is his chance of escaping the cliff?
From the book that I read, a solution involves proving continuity of some function. Here's a relevant portion of the solution:
What I don't understand is, why continuity comes into play here? Solution from other sources (example) that also involve solving for P1 don't mention continuity at all. It seems to me that the author really stresses the importance of continuity, whose proof he skips by the way ("beyond the scope of this book").
probability continuity random-walk
edited Jul 24 at 15:38
Mike Pierce
11k93574
asked Jul 23 at 4:38
Chav Likit
495
Perhaps the author is concerned about taking the square root of $P_1$? I'm honestly not sure - the author seems to be overly pedantic about the continuity here, in my opinion at least
– Brevan Ellefsen
Jul 23 at 5:08
3
You begin with "for all $p$, $P_1=1$ or $P_1=(1-p)/p$; for $p=1$, $P_1=0$; for $p in [0,1/2]$, $P_1=1$". Then the switch must occur at some $p in [1/2,1]$ which must be $1/2$ in order to ensure continuity. But this approach is not the only way, and indeed the way I would usually do it is to pass from a finite version of the problem (with an additional boundary condition) to the infinite one.
– Ian
Jul 23 at 5:33
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I have trouble understanding solution of the Drunkard's walk problem. Here is the original statement of the problem:
There once was a drunk man who wandered far too close to a cliff. From where he stands, one step forward would send the drunk man over the edge. He takes random steps, either towards or away from the cliff. At any step, his probability of taking a step away is 2/3 and a step towards the cliff is 1/3.
What is his chance of escaping the cliff?
From the book that I read, a solution involves proving continuity of some function. Here's a relevant portion of the solution:
What I don't understand is, why continuity comes into play here? Solution from other sources (example) that also involve solving for P1 don't mention continuity at all. It seems to me that the author really stresses the importance of continuity, whose proof he skips by the way ("beyond the scope of this book").
probability continuity random-walk
edited Jul 24 at 15:38
Mike Pierce
11k93574
asked Jul 23 at 4:38
Chav Likit
495
I have trouble understanding solution of the Drunkard's walk problem. Here is the original statement of the problem:
There once was a drunk man who wandered far too close to a cliff. From where he stands, one step forward would send the drunk man over the edge. He takes random steps, either towards or away from the cliff. At any step, his probability of taking a step away is 2/3 and a step towards the cliff is 1/3.
What is his chance of escaping the cliff?
From the book that I read, a solution involves proving continuity of some function. Here's a relevant portion of the solution:
What I don't understand is, why continuity comes into play here? Solution from other sources (example) that also involve solving for P1 don't mention continuity at all. It seems to me that the author really stresses the importance of continuity, whose proof he skips by the way ("beyond the scope of this book").
probability continuity random-walk
edited Jul 24 at 15:38
Mike Pierce
11k93574
asked Jul 23 at 4:38
Chav Likit
495
edited Jul 24 at 15:38
Mike Pierce
11k93574
edited Jul 24 at 15:38
Mike Pierce
11k93574
edited Jul 24 at 15:38
Mike Pierce
11k93574
11k93574
asked Jul 23 at 4:38
Chav Likit
495
asked Jul 23 at 4:38
Chav Likit
495
asked Jul 23 at 4:38
Chav Likit
495
495
Perhaps the author is concerned about taking the square root of $P_1$? I'm honestly not sure - the author seems to be overly pedantic about the continuity here, in my opinion at least
– Brevan Ellefsen
Jul 23 at 5:08
3
You begin with "for all $p$, $P_1=1$ or $P_1=(1-p)/p$; for $p=1$, $P_1=0$; for $p in [0,1/2]$, $P_1=1$". Then the switch must occur at some $p in [1/2,1]$ which must be $1/2$ in order to ensure continuity. But this approach is not the only way, and indeed the way I would usually do it is to pass from a finite version of the problem (with an additional boundary condition) to the infinite one.
– Ian
Jul 23 at 5:33
add a comment |Â
Perhaps the author is concerned about taking the square root of $P_1$? I'm honestly not sure - the author seems to be overly pedantic about the continuity here, in my opinion at least
– Brevan Ellefsen
Jul 23 at 5:08
3
You begin with "for all $p$, $P_1=1$ or $P_1=(1-p)/p$; for $p=1$, $P_1=0$; for $p in [0,1/2]$, $P_1=1$". Then the switch must occur at some $p in [1/2,1]$ which must be $1/2$ in order to ensure continuity. But this approach is not the only way, and indeed the way I would usually do it is to pass from a finite version of the problem (with an additional boundary condition) to the infinite one.
– Ian
Jul 23 at 5:33
Perhaps the author is concerned about taking the square root of $P_1$? I'm honestly not sure - the author seems to be overly pedantic about the continuity here, in my opinion at least
– Brevan Ellefsen
Jul 23 at 5:08
Perhaps the author is concerned about taking the square root of $P_1$? I'm honestly not sure - the author seems to be overly pedantic about the continuity here, in my opinion at least
– Brevan Ellefsen
Jul 23 at 5:08
3
3
You begin with "for all $p$, $P_1=1$ or $P_1=(1-p)/p$; for $p=1$, $P_1=0$; for $p in [0,1/2]$, $P_1=1$". Then the switch must occur at some $p in [1/2,1]$ which must be $1/2$ in order to ensure continuity. But this approach is not the only way, and indeed the way I would usually do it is to pass from a finite version of the problem (with an additional boundary condition) to the infinite one.
– Ian
Jul 23 at 5:33
You begin with "for all $p$, $P_1=1$ or $P_1=(1-p)/p$; for $p=1$, $P_1=0$; for $p in [0,1/2]$, $P_1=1$". Then the switch must occur at some $p in [1/2,1]$ which must be $1/2$ in order to ensure continuity. But this approach is not the only way, and indeed the way I would usually do it is to pass from a finite version of the problem (with an additional boundary condition) to the infinite one.
– Ian
Jul 23 at 5:33
add a comment |Â
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Perhaps the author is concerned about taking the square root of $P_1$? I'm honestly not sure - the author seems to be overly pedantic about the continuity here, in my opinion at least
– Brevan Ellefsen
Jul 23 at 5:08
3
You begin with "for all $p$, $P_1=1$ or $P_1=(1-p)/p$; for $p=1$, $P_1=0$; for $p in [0,1/2]$, $P_1=1$". Then the switch must occur at some $p in [1/2,1]$ which must be $1/2$ in order to ensure continuity. But this approach is not the only way, and indeed the way I would usually do it is to pass from a finite version of the problem (with an additional boundary condition) to the infinite one.
– Ian
Jul 23 at 5:33