Another intuitive proof of Central Limit Theorem [closed]

Clash Royale CLAN TAG#URR8PPP

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if one prove that the distribution of the sample mean is symmetrical, and prove that its mean is the mean of the original population and it's variance is $fracsigma^2N$, is it possible to prove the CLT arguing that the only PDF with these properties has a normal distribution ?

probability statistics

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edited Jul 31 at 18:39

asked Jul 31 at 18:33

Koinos

485

closed as unclear what you're asking by Marcus M, Did, amWhy, Isaac Browne, Xander Henderson Aug 1 at 2:12

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Well, that's not true, so...
  – Qiaochu Yuan
  Jul 31 at 18:37
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  I'm sorry i edited
  – Koinos
  Jul 31 at 18:39
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  That's still not true. There are many distributions with those properties.
  – Qiaochu Yuan
  Jul 31 at 18:40
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  And if i also prove that is monotonic on the left and right sides of the mean ?
  – Koinos
  Jul 31 at 18:43
  

  
  
  
  

add a comment | 

up vote
0
down vote

favorite

if one prove that the distribution of the sample mean is symmetrical, and prove that its mean is the mean of the original population and it's variance is $fracsigma^2N$, is it possible to prove the CLT arguing that the only PDF with these properties has a normal distribution ?

probability statistics

share|cite|improve this question

edited Jul 31 at 18:39

asked Jul 31 at 18:33

Koinos

485

closed as unclear what you're asking by Marcus M, Did, amWhy, Isaac Browne, Xander Henderson Aug 1 at 2:12

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Well, that's not true, so...
  – Qiaochu Yuan
  Jul 31 at 18:37
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  I'm sorry i edited
  – Koinos
  Jul 31 at 18:39
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  That's still not true. There are many distributions with those properties.
  – Qiaochu Yuan
  Jul 31 at 18:40
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  And if i also prove that is monotonic on the left and right sides of the mean ?
  – Koinos
  Jul 31 at 18:43
  

  
  
  
  

add a comment | 

up vote
0
down vote

favorite

up vote
0
down vote

favorite

if one prove that the distribution of the sample mean is symmetrical, and prove that its mean is the mean of the original population and it's variance is $fracsigma^2N$, is it possible to prove the CLT arguing that the only PDF with these properties has a normal distribution ?

probability statistics

share|cite|improve this question

edited Jul 31 at 18:39

asked Jul 31 at 18:33

Koinos

485

if one prove that the distribution of the sample mean is symmetrical, and prove that its mean is the mean of the original population and it's variance is $fracsigma^2N$, is it possible to prove the CLT arguing that the only PDF with these properties has a normal distribution ?

probability statistics

share|cite|improve this question

edited Jul 31 at 18:39

asked Jul 31 at 18:33

Koinos

485

share|cite|improve this question

share|cite|improve this question

edited Jul 31 at 18:39

edited Jul 31 at 18:39

edited Jul 31 at 18:39

asked Jul 31 at 18:33

Koinos

485

asked Jul 31 at 18:33

Koinos

485

asked Jul 31 at 18:33

Koinos

485

485

closed as unclear what you're asking by Marcus M, Did, amWhy, Isaac Browne, Xander Henderson Aug 1 at 2:12

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

closed as unclear what you're asking by Marcus M, Did, amWhy, Isaac Browne, Xander Henderson Aug 1 at 2:12

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Well, that's not true, so...
  – Qiaochu Yuan
  Jul 31 at 18:37
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  I'm sorry i edited
  – Koinos
  Jul 31 at 18:39
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  That's still not true. There are many distributions with those properties.
  – Qiaochu Yuan
  Jul 31 at 18:40
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  And if i also prove that is monotonic on the left and right sides of the mean ?
  – Koinos
  Jul 31 at 18:43
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Well, that's not true, so...
  – Qiaochu Yuan
  Jul 31 at 18:37
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  I'm sorry i edited
  – Koinos
  Jul 31 at 18:39
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  That's still not true. There are many distributions with those properties.
  – Qiaochu Yuan
  Jul 31 at 18:40
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  And if i also prove that is monotonic on the left and right sides of the mean ?
  – Koinos
  Jul 31 at 18:43
  

  
  
  
  

Well, that's not true, so...
– Qiaochu Yuan
Jul 31 at 18:37

Well, that's not true, so...
– Qiaochu Yuan
Jul 31 at 18:37

I'm sorry i edited
– Koinos
Jul 31 at 18:39

I'm sorry i edited
– Koinos
Jul 31 at 18:39

1

1

That's still not true. There are many distributions with those properties.
– Qiaochu Yuan
Jul 31 at 18:40

That's still not true. There are many distributions with those properties.
– Qiaochu Yuan
Jul 31 at 18:40

And if i also prove that is monotonic on the left and right sides of the mean ?
– Koinos
Jul 31 at 18:43

And if i also prove that is monotonic on the left and right sides of the mean ?
– Koinos
Jul 31 at 18:43

add a comment | 

1 Answer
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oldest

votes

up vote
1
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As noted in the comments, no that's not enough. Furthermore, the sample mean need not be symmetrical if the underlying distribution of the samples isn't. However, you are coming close to another way of proving the central limit theorem: it's not enough to just show that the sample mean has the correct mean and variance; however, if you show that all moments of the (recentered and renormalized) sample mean converge to the moments of the standard normal, then you get a proof of the central limit theorem. This is known as the moment method or the method of moments, although "the method of moments" also refers to a different concept in statistics. For a proof of the CLT using the moment method, see this blog post by Terry Tao.

share|cite|improve this answer

answered Jul 31 at 18:43

Marcus M

8,1631847

• 
  
  
  

  
  

  
  
  
  
  
  

  
  However it seems not an intuitive proof :(
  – Koinos
  Jul 31 at 18:47
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  It's unclear what you're looking for; there are a lot of proofs of the central limit theorem, but it's hard to say if any are intuitive. One that's somewhat intuitive but quite difficult to carry out a proof of is by noting that the entropy of the sample means increases with $n$, and that the standard normal maximizes entropy. Here's a thread on MathOverflow about that.
  – Marcus M
  Jul 31 at 18:54
  

  
  
  
  

add a comment | 

1 Answer
1

active

oldest

votes

1 Answer
1

active

oldest

votes

active

oldest

votes

active

oldest

votes

up vote
1
down vote

As noted in the comments, no that's not enough. Furthermore, the sample mean need not be symmetrical if the underlying distribution of the samples isn't. However, you are coming close to another way of proving the central limit theorem: it's not enough to just show that the sample mean has the correct mean and variance; however, if you show that all moments of the (recentered and renormalized) sample mean converge to the moments of the standard normal, then you get a proof of the central limit theorem. This is known as the moment method or the method of moments, although "the method of moments" also refers to a different concept in statistics. For a proof of the CLT using the moment method, see this blog post by Terry Tao.

share|cite|improve this answer

answered Jul 31 at 18:43

Marcus M

8,1631847

• 
  
  
  

  
  

  
  
  
  
  
  

  
  However it seems not an intuitive proof :(
  – Koinos
  Jul 31 at 18:47
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  It's unclear what you're looking for; there are a lot of proofs of the central limit theorem, but it's hard to say if any are intuitive. One that's somewhat intuitive but quite difficult to carry out a proof of is by noting that the entropy of the sample means increases with $n$, and that the standard normal maximizes entropy. Here's a thread on MathOverflow about that.
  – Marcus M
  Jul 31 at 18:54
  

  
  
  
  

add a comment | 

up vote
1
down vote

As noted in the comments, no that's not enough. Furthermore, the sample mean need not be symmetrical if the underlying distribution of the samples isn't. However, you are coming close to another way of proving the central limit theorem: it's not enough to just show that the sample mean has the correct mean and variance; however, if you show that all moments of the (recentered and renormalized) sample mean converge to the moments of the standard normal, then you get a proof of the central limit theorem. This is known as the moment method or the method of moments, although "the method of moments" also refers to a different concept in statistics. For a proof of the CLT using the moment method, see this blog post by Terry Tao.

share|cite|improve this answer

answered Jul 31 at 18:43

Marcus M

8,1631847

• 
  
  
  

  
  

  
  
  
  
  
  

  
  However it seems not an intuitive proof :(
  – Koinos
  Jul 31 at 18:47
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  It's unclear what you're looking for; there are a lot of proofs of the central limit theorem, but it's hard to say if any are intuitive. One that's somewhat intuitive but quite difficult to carry out a proof of is by noting that the entropy of the sample means increases with $n$, and that the standard normal maximizes entropy. Here's a thread on MathOverflow about that.
  – Marcus M
  Jul 31 at 18:54
  

  
  
  
  

add a comment | 

up vote
1
down vote

up vote
1
down vote

As noted in the comments, no that's not enough. Furthermore, the sample mean need not be symmetrical if the underlying distribution of the samples isn't. However, you are coming close to another way of proving the central limit theorem: it's not enough to just show that the sample mean has the correct mean and variance; however, if you show that all moments of the (recentered and renormalized) sample mean converge to the moments of the standard normal, then you get a proof of the central limit theorem. This is known as the moment method or the method of moments, although "the method of moments" also refers to a different concept in statistics. For a proof of the CLT using the moment method, see this blog post by Terry Tao.

share|cite|improve this answer

answered Jul 31 at 18:43

Marcus M

8,1631847

As noted in the comments, no that's not enough. Furthermore, the sample mean need not be symmetrical if the underlying distribution of the samples isn't. However, you are coming close to another way of proving the central limit theorem: it's not enough to just show that the sample mean has the correct mean and variance; however, if you show that all moments of the (recentered and renormalized) sample mean converge to the moments of the standard normal, then you get a proof of the central limit theorem. This is known as the moment method or the method of moments, although "the method of moments" also refers to a different concept in statistics. For a proof of the CLT using the moment method, see this blog post by Terry Tao.

share|cite|improve this answer

answered Jul 31 at 18:43

Marcus M

8,1631847

share|cite|improve this answer

share|cite|improve this answer

answered Jul 31 at 18:43

Marcus M

8,1631847

answered Jul 31 at 18:43

Marcus M

8,1631847

answered Jul 31 at 18:43

Marcus M

8,1631847

8,1631847

• 
  
  
  

  
  

  
  
  
  
  
  

  
  However it seems not an intuitive proof :(
  – Koinos
  Jul 31 at 18:47
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  It's unclear what you're looking for; there are a lot of proofs of the central limit theorem, but it's hard to say if any are intuitive. One that's somewhat intuitive but quite difficult to carry out a proof of is by noting that the entropy of the sample means increases with $n$, and that the standard normal maximizes entropy. Here's a thread on MathOverflow about that.
  – Marcus M
  Jul 31 at 18:54
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  

  
  
  
  
  
  

  
  However it seems not an intuitive proof :(
  – Koinos
  Jul 31 at 18:47
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  It's unclear what you're looking for; there are a lot of proofs of the central limit theorem, but it's hard to say if any are intuitive. One that's somewhat intuitive but quite difficult to carry out a proof of is by noting that the entropy of the sample means increases with $n$, and that the standard normal maximizes entropy. Here's a thread on MathOverflow about that.
  – Marcus M
  Jul 31 at 18:54
  

  
  
  
  

However it seems not an intuitive proof :(
– Koinos
Jul 31 at 18:47

However it seems not an intuitive proof :(
– Koinos
Jul 31 at 18:47

1

1

It's unclear what you're looking for; there are a lot of proofs of the central limit theorem, but it's hard to say if any are intuitive. One that's somewhat intuitive but quite difficult to carry out a proof of is by noting that the entropy of the sample means increases with $n$, and that the standard normal maximizes entropy. Here's a thread on MathOverflow about that.
– Marcus M
Jul 31 at 18:54

It's unclear what you're looking for; there are a lot of proofs of the central limit theorem, but it's hard to say if any are intuitive. One that's somewhat intuitive but quite difficult to carry out a proof of is by noting that the entropy of the sample means increases with $n$, and that the standard normal maximizes entropy. Here's a thread on MathOverflow about that.
– Marcus M
Jul 31 at 18:54

add a comment |