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Author Topic:   On the proportion of Nucleotides in the Genome and what it can tell us about Evolutio
Dr Adequate
Member (Idle past 284 days)
Posts: 16113
Joined: 07-20-2006


Message 16 of 61 (524363)
09-16-2009 8:03 AM
Reply to: Message 14 by Peepul
09-16-2009 6:06 AM


Re: Statistics --- ur doin it rong
I think it does Dr A - the expectation value of the excess of heads or tails is the square root of n, where n is the number of tosses
My emphasis.
Do we not expect any particular random walk to meet the heads = tails axis infinitely many times as n tends to infinity?
Especially bearing in mind that this particular random walk has a floor and a ceiling.
But thanks for telling me that, 'cos I'd been wondering what the function was. Do you have a reference? But I'm not sure that it's relevant to my point.
Edited by Dr Adequate, : No reason given.
Edited by Dr Adequate, : No reason given.

This message is a reply to:
 Message 14 by Peepul, posted 09-16-2009 6:06 AM Peepul has not replied

Replies to this message:
 Message 17 by Dr Jack, posted 09-16-2009 8:17 AM Dr Adequate has replied
 Message 28 by Dr Jack, posted 09-16-2009 11:01 AM Dr Adequate has replied

  
Dr Jack
Member
Posts: 3514
From: Immigrant in the land of Deutsch
Joined: 07-14-2003
Member Rating: 8.7


Message 17 of 61 (524364)
09-16-2009 8:17 AM
Reply to: Message 16 by Dr Adequate
09-16-2009 8:03 AM


Re: Statistics --- ur doin it rong
The median modal number of times for a sequence of heads and tails to cross the zero line is 0, with all higher numbers occur with decreasing probibility.
The reason for this is quite obvious if you think about it: because all future flips occur independently of those flips that have already happened, the chances are that there will not be a trend reversing any initial discrepency from zero.
Edited by Mr Jack, : Added explaination
Edited by Mr Jack, : Meant mode not median

This message is a reply to:
 Message 16 by Dr Adequate, posted 09-16-2009 8:03 AM Dr Adequate has replied

Replies to this message:
 Message 18 by Dr Adequate, posted 09-16-2009 8:42 AM Dr Jack has replied

  
Dr Adequate
Member (Idle past 284 days)
Posts: 16113
Joined: 07-20-2006


Message 18 of 61 (524369)
09-16-2009 8:42 AM
Reply to: Message 17 by Dr Jack
09-16-2009 8:17 AM


Re: Statistics --- ur doin it rong
The median number of times for a sequence of heads and tails to cross the zero line is 0, with all higher numbers occur with decreasing probibility.
How can zero be the median?
The reason for this is quite obvious if you think about it: because all future flips occur independently of those flips that have already happened, the chances are that there will not be a trend reversing any initial discrepency from zero.
No trend, certainly.
What of it?
---
I think you're still confusing the average difference from zero as n tends to infinity with the path of a particular random walk.
---
In any case, we are here discussing a random walk with a floor and a ceiling.

This message is a reply to:
 Message 17 by Dr Jack, posted 09-16-2009 8:17 AM Dr Jack has replied

Replies to this message:
 Message 19 by jacortina, posted 09-16-2009 9:21 AM Dr Adequate has replied
 Message 20 by Dr Jack, posted 09-16-2009 9:59 AM Dr Adequate has replied

  
jacortina
Member (Idle past 5083 days)
Posts: 64
Joined: 08-07-2009


Message 19 of 61 (524375)
09-16-2009 9:21 AM
Reply to: Message 18 by Dr Adequate
09-16-2009 8:42 AM


Re: Statistics --- ur doin it rong
How can zero be the median?
Talking about the median in this context actually means (the way I see it) comparing multiple sequences of coin flipping. If, in each of five sequences, the number of times the zero-difference point is crossed are (0,0,0,2,5), the median is zero. With six sequences (0,0,0,0,3,4), the median is also zero.
Zero can certainly be the middle number of an ordered set with an odd number of members or the average of the two middle numbers of an ordered set with an even number of members.
Basically, claiming a zero median is claiming that more than half the time, a sequence will NOT cross the zero-difference line.

This message is a reply to:
 Message 18 by Dr Adequate, posted 09-16-2009 8:42 AM Dr Adequate has replied

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Dr Jack
Member
Posts: 3514
From: Immigrant in the land of Deutsch
Joined: 07-14-2003
Member Rating: 8.7


Message 20 of 61 (524380)
09-16-2009 9:59 AM
Reply to: Message 18 by Dr Adequate
09-16-2009 8:42 AM


Re: Statistics --- ur doin it rong
How can zero be the median?
Sorry, I meant mode. My bad.
I think you're still confusing the average difference from zero as n tends to infinity with the path of a particular random walk.
No, I'm not. More paths will not ever cross the zero line than any other particular number of crosses; what is more as the number of crosses increases, the number of paths that have that number of crosses decreases.

This message is a reply to:
 Message 18 by Dr Adequate, posted 09-16-2009 8:42 AM Dr Adequate has replied

Replies to this message:
 Message 24 by Dr Adequate, posted 09-16-2009 10:44 AM Dr Jack has replied

  
Dr Adequate
Member (Idle past 284 days)
Posts: 16113
Joined: 07-20-2006


Message 21 of 61 (524393)
09-16-2009 10:37 AM
Reply to: Message 19 by jacortina
09-16-2009 9:21 AM


Re: Statistics --- ur doin it rong
Talking about the median in this context actually means (the way I see it) comparing multiple sequences of coin flipping.
One might call it the "mean median".
If, in each of five sequences, the number of times the zero-difference point is crossed are (0,0,0,2,5), the median is zero. With six sequences (0,0,0,0,3,4), the median is also zero.
Zero can certainly be the middle number of an ordered set with an odd number of members or the average of the two middle numbers of an ordered set with an even number of members.
Basically, claiming a zero median is claiming that more than half the time, a sequence will NOT cross the zero-difference line.
How can zero be the median?
Talking about the median in this context actually means (the way I see it) comparing multiple sequences of coin flipping. If, in each of five sequences, the number of times the zero-difference point is crossed are (0,0,0,2,5), the median is zero. With six sequences (0,0,0,0,3,4), the median is also zero.
Zero can certainly be the middle number of an ordered set with an odd number of members or the average of the two middle numbers of an ordered set with an even number of members.
Basically, claiming a zero median is claiming that more than half the time, a sequence will NOT cross the zero-difference line.
Right, that's exactly what is being claimed. Now let's think this through.
In the first two tosses of the coin, we will either get HH or TT, in which case we will not meet the h = t axis --- or we will get HT or TH, in which case we will meet the h = t axis.
So after just the first two flips of the coin, it's fifty-fifty that we shall have met the h = t axis once.
Now, further flips of the coin can only increase, not reduce, the probability that we meet the h = t axis at least once.
Therefore, for n > 2, the "mean median" cannot be zero.

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Replies to this message:
 Message 22 by NosyNed, posted 09-16-2009 10:42 AM Dr Adequate has replied
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NosyNed
Member
Posts: 8996
From: Canada
Joined: 04-04-2003


Message 22 of 61 (524394)
09-16-2009 10:42 AM
Reply to: Message 21 by Dr Adequate
09-16-2009 10:37 AM


Re: Statistics --- ur doin it rong
So after just the first two flips of the coin, it's fifty-fifty that we shall have met the h = t axis once.
Meeting is not crossing. In these examples the crossings are zero.
(presuming I understand what is being put forward)

This message is a reply to:
 Message 21 by Dr Adequate, posted 09-16-2009 10:37 AM Dr Adequate has replied

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Dr Jack
Member
Posts: 3514
From: Immigrant in the land of Deutsch
Joined: 07-14-2003
Member Rating: 8.7


Message 23 of 61 (524395)
09-16-2009 10:44 AM
Reply to: Message 21 by Dr Adequate
09-16-2009 10:37 AM


Re: Statistics --- ur doin it rong
Nosy, it correct, I was discussing crossings, not meetings.
And, as I said, I meant mode not median. Again, my bad.

This message is a reply to:
 Message 21 by Dr Adequate, posted 09-16-2009 10:37 AM Dr Adequate has not replied

  
Dr Adequate
Member (Idle past 284 days)
Posts: 16113
Joined: 07-20-2006


Message 24 of 61 (524396)
09-16-2009 10:44 AM
Reply to: Message 20 by Dr Jack
09-16-2009 9:59 AM


Re: Statistics --- ur doin it rong
Sorry, I meant mode. My bad.
Oh, now you're going to bring the mode into this?
Apart from wondering why, I'd like to see your reasoning.
No, I'm not. More paths will not ever cross the zero line than any other particular number of crosses; what is more as the number of crosses increases, the number of paths that have that number of crosses decreases.
I didn't follow that.
Let me talk about the mean. You remember that, the thing we use when doing statistics and talking about expected values?
Well, rather than figure it out from first principles, I just got my computer to toss lots of coins for me.
Each number in the right hand column is the mean of a thousand trials.
number of coin tosses    number of times heads - tails = 0

2:                         0.500
4:                         0.853
8:                         1.438
16:                        2.360
32:                        3.644
64:                        5.165
128:                       8.226
256:                      11.866
512:                      17.422
1024:                     23.681
2048:                     34.958
4096:                     52.290
8192:                     72.220
16384:                   101.483
32768:                   139.231
65536:                   201.394
131072:                  275.738
262144:                  420.173
524288:                  560.344
1048576:                 826.211
I therefore stand by my claim that the number of times we'll meet the h = t axis will indeed tend to infinity with the number of coin tosses.
If you would like to dispute that, perhaps you could try thinking for a few moments about where, in that case, the asymptote would be.
Edited by Dr Adequate, : No reason given.
Edited by Dr Adequate, : No reason given.
Edited by Dr Adequate, : No reason given.

This message is a reply to:
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Replies to this message:
 Message 26 by Dr Jack, posted 09-16-2009 10:51 AM Dr Adequate has replied
 Message 49 by Richard Townsend, posted 09-17-2009 7:05 PM Dr Adequate has replied

  
Dr Adequate
Member (Idle past 284 days)
Posts: 16113
Joined: 07-20-2006


Message 25 of 61 (524397)
09-16-2009 10:50 AM
Reply to: Message 22 by NosyNed
09-16-2009 10:42 AM


Re: Statistics --- ur doin it rong
Meeting is not crossing. In these examples the crossings are zero.
(presuming I understand what is being put forward)
Apparently not. If the number of heads minus the number of tails is zero, then we have indeed met the h = t axis.

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Dr Jack
Member
Posts: 3514
From: Immigrant in the land of Deutsch
Joined: 07-14-2003
Member Rating: 8.7


Message 26 of 61 (524398)
09-16-2009 10:51 AM
Reply to: Message 24 by Dr Adequate
09-16-2009 10:44 AM


Re: Statistics --- ur doin it rong
Mean is an extremely poor choice for a situation such as this - especially when considering the infinite limit - because of the large value distortion effect, and the non-normal distribution of the probabilities.
The most probable number of crosses is 0, followed by 1, followed by 2, followed by 3, etc.

This message is a reply to:
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Dr Jack
Member
Posts: 3514
From: Immigrant in the land of Deutsch
Joined: 07-14-2003
Member Rating: 8.7


Message 27 of 61 (524399)
09-16-2009 10:53 AM
Reply to: Message 25 by Dr Adequate
09-16-2009 10:50 AM


Re: Statistics --- ur doin it rong
But you've not crossed it unless the next toss is the same.
HTH meets the zero line, it does not cross it
HTT meets and then crosses the zero line.
Of the possible 3 toss sequences, then, HTT and THH cross the line once; the other six do not cross it at all.

This message is a reply to:
 Message 25 by Dr Adequate, posted 09-16-2009 10:50 AM Dr Adequate has replied

Replies to this message:
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Dr Jack
Member
Posts: 3514
From: Immigrant in the land of Deutsch
Joined: 07-14-2003
Member Rating: 8.7


Message 28 of 61 (524402)
09-16-2009 11:01 AM
Reply to: Message 16 by Dr Adequate
09-16-2009 8:03 AM


Re: Statistics --- ur doin it rong
After looking it up, I realise I have erred. In fact, the result I stated holds for a finite random walk, but it does not hold for an infinite random walk.

This message is a reply to:
 Message 16 by Dr Adequate, posted 09-16-2009 8:03 AM Dr Adequate has replied

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Dr Adequate
Member (Idle past 284 days)
Posts: 16113
Joined: 07-20-2006


Message 29 of 61 (524403)
09-16-2009 11:01 AM
Reply to: Message 27 by Dr Jack
09-16-2009 10:53 AM


Re: Statistics --- ur doin it rong
But you've not crossed it unless the next toss is the same.
So, you are not merely wrong, you're also calculating the wrong thing. Wrongly.
* facepalm *
Edited by Dr Adequate, : No reason given.

This message is a reply to:
 Message 27 by Dr Jack, posted 09-16-2009 10:53 AM Dr Jack has replied

Replies to this message:
 Message 31 by Dr Jack, posted 09-16-2009 11:05 AM Dr Adequate has replied

  
Dr Adequate
Member (Idle past 284 days)
Posts: 16113
Joined: 07-20-2006


Message 30 of 61 (524405)
09-16-2009 11:04 AM
Reply to: Message 26 by Dr Jack
09-16-2009 10:51 AM


Re: Statistics --- ur doin it rong
Mean is an extremely poor choice for a situation such as this - especially when considering the infinite limit - because of the large value distortion effect, and the non-normal distribution of the probabilities.
The most probable number of crosses is 0, followed by 1, followed by 2, followed by 3, etc.
* jaw drops *
I'm going to guess that you're not a mathematician.

This message is a reply to:
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