# Recursive distribution experiment

## Will this distribution look binomial?

Poll closed Jul 26, 2009.

4 vote(s)
21.1%

4 vote(s)
21.1%

2 vote(s)
10.5%

0 vote(s)
0.0%

3 vote(s)
15.8%

1 vote(s)
5.3%

0 vote(s)
0.0%

2 vote(s)
10.5%

2 vote(s)
10.5%

0 vote(s)
0.0%
11. ### I would bet my life on it.

1 vote(s)
5.3%
1. Jul 19, 2009

### junglebeast

If you consider the results of this poll as the probability mass function for a discrete distribution, do you think it will approximate a binomial distribution?

Think about it...but do not look at the results before casting your vote.

2. Jul 19, 2009

### AUMathTutor

Interesting idea. I'll be interested to see how this develops.

Can we discuss it in the thread, or should we keep our reasoning to ourselves until more people have responded?

3. Jul 19, 2009

### junglebeast

I was inspired by a slashdot poll which appeared to have an exponential distribution. I thought to myself, hmph..I wish the question of this poll was, "what distribution will this poll have?"

It took me a while to think of a way to create a poll that might have a similar effect..where people's vote is driving the actual distribution. Do you think it will work?

Feel free to discuss :)

4. Jul 19, 2009

### AUMathTutor

I think that it's better to have a range of values that sort of span a quantitative spectrum. I'm not sure a list of qualitatively different options would work.

Well, if people just put down a random answer, you would expect a uniform distribution. However, that's not the exercise.

If you ask me, people will reason like this:
(1) It's only binomial if you put a bunch towards the middle, and putting them in the middle is saying you don't really know what a binomial is or how it will work out. So this isn't a rationally consistent choice.
(2) For the same reason, loading them up at the end of the list doesn't make much sense, because then you'll be saying you're sure it will be binomial when you're not helping it towards that end. And if you don't, why should others?
(3) Putting it at the front, you're saying that it won't end up being binomial, and in fact your choice helps get it away from a binomial. Assuming other people think the same way, they will also be helping your interpretation. so this is the only logical choice.

Therefore, I would expect that the distribution would be roughly exponential.

5. Jul 20, 2009

### junglebeast

The values range from 0-10. Actually there are 11 possible answers so that there is an exact middle to eliminate bias. The named values just correspond to 0 = zero probability, 1 = 100% probability (in your view).

If there is an underlying probability of people that believe it will be binomial (like 20%) and they do not look at current poll results before voting, and they vote honestly, then this should approximate a binomial distribution if they place their vote according to the probability that they think it has.

For example, if you think it is 50% probability then you cast your vote exactly in the middle. Because the probability in each discrete interval is a sum (non-normalized), I think this should work.

6. Jul 20, 2009

### AUMathTutor

But the problem with that is that voting in the middle isn't a logical "fixed point" for this poll.

If you vote in the middle, then you logically assume others would as well. But if everybody voted in the middle, then you would retroactively change your vote to be a higher number. You don't have to see actual results to do this logic.

If you vote near the top, you would assume others would logically do so as well. However, if this is the case, you would retroactively change your vote to being a low number, since a high-skewed distribution is not binomial.

If you vote a low number, you would assume others would as well. And this would lead to a situation not resembling a binomial, and therefore you would not need to go back and change your answer; your answer would be right, if everybody else did the same (logical) thing.

It's a very Kantian argument, but there you go. I think that if the results don't look skewed towards the top, I'd be curious to know what logical framework others are using to base their guesses on.

If people are just picking a percentage that sounds good, that's essentially random, and will probably end up looking binomial just because people will shy away from extremes and shoot for a middle value. If people weren't biased, it would just like the random experiment I hypothesized and would look practically uniform.

7. Jul 20, 2009

### junglebeast

You make a logical argument. However, not everyone will give it this much thought...so we will see :)