Fixed Point in a random positive integer function
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I was talking with a friend this morning and he asked me this problem:
For a random positive integer function ( from N to N) What is the possibility that the function contains a fixed point, i.e. f(x)=x
Without knowledge of measure theory, I do not think the problem is well defined, though it is supposedly an “equal chance” model.
The problem with this definition is that, if we consider functions of set {1,2,...,n} to N we can inductively get a possibility of 0 ( Not so sure, here is what I did: say g(n) is the possibility from all functions from set 1 to n, base case g(1)=0, and of g(n)=0, g(n+1)=g(n)+g(1)=0
Yet similarly, if we consider the functions from {1,...,n} to {1 to n} we will get 1/e eventually.
Can someone point out flaw in each reasoning?
Merry Christmas.
real-analysis probability-theory
asked Dec 25 '18 at 15:50
Yuan LiaoYuan Liao
31
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add a comment |
$begingroup$
I was talking with a friend this morning and he asked me this problem:
For a random positive integer function ( from N to N) What is the possibility that the function contains a fixed point, i.e. f(x)=x
Without knowledge of measure theory, I do not think the problem is well defined, though it is supposedly an “equal chance” model.
The problem with this definition is that, if we consider functions of set {1,2,...,n} to N we can inductively get a possibility of 0 ( Not so sure, here is what I did: say g(n) is the possibility from all functions from set 1 to n, base case g(1)=0, and of g(n)=0, g(n+1)=g(n)+g(1)=0
Yet similarly, if we consider the functions from {1,...,n} to {1 to n} we will get 1/e eventually.
Can someone point out flaw in each reasoning?
Merry Christmas.
real-analysis probability-theory
asked Dec 25 '18 at 15:50
Yuan LiaoYuan Liao
31
$endgroup$
add a comment |
$begingroup$
I was talking with a friend this morning and he asked me this problem:
For a random positive integer function ( from N to N) What is the possibility that the function contains a fixed point, i.e. f(x)=x
Without knowledge of measure theory, I do not think the problem is well defined, though it is supposedly an “equal chance” model.
The problem with this definition is that, if we consider functions of set {1,2,...,n} to N we can inductively get a possibility of 0 ( Not so sure, here is what I did: say g(n) is the possibility from all functions from set 1 to n, base case g(1)=0, and of g(n)=0, g(n+1)=g(n)+g(1)=0
Yet similarly, if we consider the functions from {1,...,n} to {1 to n} we will get 1/e eventually.
Can someone point out flaw in each reasoning?
Merry Christmas.
real-analysis probability-theory
asked Dec 25 '18 at 15:50
Yuan LiaoYuan Liao
31
$endgroup$
I was talking with a friend this morning and he asked me this problem:
For a random positive integer function ( from N to N) What is the possibility that the function contains a fixed point, i.e. f(x)=x
Without knowledge of measure theory, I do not think the problem is well defined, though it is supposedly an “equal chance” model.
The problem with this definition is that, if we consider functions of set {1,2,...,n} to N we can inductively get a possibility of 0 ( Not so sure, here is what I did: say g(n) is the possibility from all functions from set 1 to n, base case g(1)=0, and of g(n)=0, g(n+1)=g(n)+g(1)=0
Yet similarly, if we consider the functions from {1,...,n} to {1 to n} we will get 1/e eventually.
Can someone point out flaw in each reasoning?
Merry Christmas.
real-analysis probability-theory
real-analysis probability-theory
asked Dec 25 '18 at 15:50
Yuan LiaoYuan Liao
31
asked Dec 25 '18 at 15:50
Yuan LiaoYuan Liao
31
asked Dec 25 '18 at 15:50
Yuan LiaoYuan Liao
31
asked Dec 25 '18 at 15:50
Yuan LiaoYuan Liao
31
asked Dec 25 '18 at 15:50
Yuan LiaoYuan Liao
31
31
add a comment |
add a comment |
1 Answer
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If $f:mathbb Ntomathbb N$ is a random function then questions arise as: what is the probability that $f(0)=n$?
Evidently it cannot be that this probability is the same for every $ninmathbb N$ so there is no uniform distribution available for $f(0)$.
That means that random functions $mathbb Ntomathbb N$ can only exist by the grace of predefined distributions that cannot be recognized as uniform.
A sort of "canonical" uniform distribution as we have on finite sets or on intervals $[a,b]$ lacks.
This makes questions like: "what is the probability that a random positive integer function has a fixed point?" senseless.
Answers on the question are bound to distributions and there are lots of them.
For different distributions we get different answers.
answered Dec 25 '18 at 16:02
drhabdrhab
101k545136
$endgroup$
$begingroup$
Hi, thanks! That is what I thought with a formal notion. After reviewing probability axioms, I might just show the conflict that no uniform distribution exists
$endgroup$
– Yuan Liao
Dec 25 '18 at 16:36
add a comment |
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$begingroup$
If $f:mathbb Ntomathbb N$ is a random function then questions arise as: what is the probability that $f(0)=n$?
Evidently it cannot be that this probability is the same for every $ninmathbb N$ so there is no uniform distribution available for $f(0)$.
That means that random functions $mathbb Ntomathbb N$ can only exist by the grace of predefined distributions that cannot be recognized as uniform.
A sort of "canonical" uniform distribution as we have on finite sets or on intervals $[a,b]$ lacks.
This makes questions like: "what is the probability that a random positive integer function has a fixed point?" senseless.
Answers on the question are bound to distributions and there are lots of them.
For different distributions we get different answers.
answered Dec 25 '18 at 16:02
drhabdrhab
101k545136
$endgroup$
$begingroup$
Hi, thanks! That is what I thought with a formal notion. After reviewing probability axioms, I might just show the conflict that no uniform distribution exists
$endgroup$
– Yuan Liao
Dec 25 '18 at 16:36
add a comment |
$begingroup$
If $f:mathbb Ntomathbb N$ is a random function then questions arise as: what is the probability that $f(0)=n$?
Evidently it cannot be that this probability is the same for every $ninmathbb N$ so there is no uniform distribution available for $f(0)$.
That means that random functions $mathbb Ntomathbb N$ can only exist by the grace of predefined distributions that cannot be recognized as uniform.
A sort of "canonical" uniform distribution as we have on finite sets or on intervals $[a,b]$ lacks.
This makes questions like: "what is the probability that a random positive integer function has a fixed point?" senseless.
Answers on the question are bound to distributions and there are lots of them.
For different distributions we get different answers.
answered Dec 25 '18 at 16:02
drhabdrhab
101k545136
$endgroup$
$begingroup$
Hi, thanks! That is what I thought with a formal notion. After reviewing probability axioms, I might just show the conflict that no uniform distribution exists
$endgroup$
– Yuan Liao
Dec 25 '18 at 16:36
add a comment |
$begingroup$
If $f:mathbb Ntomathbb N$ is a random function then questions arise as: what is the probability that $f(0)=n$?
Evidently it cannot be that this probability is the same for every $ninmathbb N$ so there is no uniform distribution available for $f(0)$.
That means that random functions $mathbb Ntomathbb N$ can only exist by the grace of predefined distributions that cannot be recognized as uniform.
A sort of "canonical" uniform distribution as we have on finite sets or on intervals $[a,b]$ lacks.
This makes questions like: "what is the probability that a random positive integer function has a fixed point?" senseless.
Answers on the question are bound to distributions and there are lots of them.
For different distributions we get different answers.
answered Dec 25 '18 at 16:02
drhabdrhab
101k545136
$endgroup$
If $f:mathbb Ntomathbb N$ is a random function then questions arise as: what is the probability that $f(0)=n$?
Evidently it cannot be that this probability is the same for every $ninmathbb N$ so there is no uniform distribution available for $f(0)$.
That means that random functions $mathbb Ntomathbb N$ can only exist by the grace of predefined distributions that cannot be recognized as uniform.
A sort of "canonical" uniform distribution as we have on finite sets or on intervals $[a,b]$ lacks.
This makes questions like: "what is the probability that a random positive integer function has a fixed point?" senseless.
Answers on the question are bound to distributions and there are lots of them.
For different distributions we get different answers.
answered Dec 25 '18 at 16:02
drhabdrhab
101k545136
edited Dec 25 '18 at 16:47
answered Dec 25 '18 at 16:02
drhabdrhab
101k545136
answered Dec 25 '18 at 16:02
drhabdrhab
101k545136
answered Dec 25 '18 at 16:02
drhabdrhab
101k545136
101k545136
$begingroup$
Hi, thanks! That is what I thought with a formal notion. After reviewing probability axioms, I might just show the conflict that no uniform distribution exists
$endgroup$
– Yuan Liao
Dec 25 '18 at 16:36
add a comment |
$begingroup$
Hi, thanks! That is what I thought with a formal notion. After reviewing probability axioms, I might just show the conflict that no uniform distribution exists
$endgroup$
– Yuan Liao
Dec 25 '18 at 16:36
$begingroup$
Hi, thanks! That is what I thought with a formal notion. After reviewing probability axioms, I might just show the conflict that no uniform distribution exists
$endgroup$
– Yuan Liao
Dec 25 '18 at 16:36
$begingroup$
Hi, thanks! That is what I thought with a formal notion. After reviewing probability axioms, I might just show the conflict that no uniform distribution exists
$endgroup$
– Yuan Liao
Dec 25 '18 at 16:36
add a comment |
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