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RolfAndreassen comments on A Request for Open Problems - Less Wrong
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"The uniform distribution centered at c" does not seem to make sense. Did you perchance mean the Gaussian distribution? Further, 'deviates' looks like jargon to me. Can we use 'samples'? I would therefore rephrase as follows, with specific example to hang one's visualisation on:
Heights of male humans are known to have a Gaussian distribution of width 10 cm around some central value <h>; unfortunately you have forgotten what the central value is. Joe is 180 cm, Stephen is 170 cm. The probability that <h> is between these two heights is 50%; explain why. Then find a better confidence interval for <h>.
I mean the continuous uniform distribution. "Centered at c" is intended to indicate that the mean of the distribution is c.
ETA: Let me be specific -- I'll use the notation of the linked Wikipedia article.
You know that b - a = 1.
c = (a + b)/2 is unknown, and the confidence interval is supposed to help you infer it.
If exactly half of all men have a height less than the central value c, than randomly picking sample will have a 50% chance of being below c. Picking two samples (A and B) results in four possible scenarios:
The interval created by (A, B) contains c in scenarios (1) and (4) and does not contain c in scenarios (2) and (3). Since each scenario has an equal chance of occurring, c is in (A, B) 50% of the time.
That is as far as I got just thinking about it. If I am on the right path I can keep plugging away.
In the Gaussian case, you can do better than (A, B) but the demonstration of that fact won't smack you in the face they way it does in the case of the uniform distribution.
One thing you can do in the uniform case is shorten the interval to at most length 1/2. Not sure if that's face-smacking enough.
You can do better than that. If the distance between the two data points is 7/4, you can shrink the 100% confidence interval to 1/4, etc. (The extreme case is as the distance between the two data points approaches 2, your 100% confidence interval approaches size zero.)
EDIT: whoops, I was stupid. Corrected 3/4 to 7/4 and 1 to 2. There, now it should be right
Do we know the heights of the men A and B? If so, we can get a better estimate of whether c lies between their heights by taking into account the difference between A and B...
That's the basic idea. Now apply it in the case of the uniform distribution.
If all men are (say) within 10 cm of each other, and the heights are uniformly distributed...
... if we have two men who are 8 cm apart, then c lies between their heights with 80% probability?
Getting there... 80% is too low.
Wait, what? It must be 100%...
That's it. The so-called 50% confidence interval sometimes contains c with certainty. Also, when x_max - x_min is much smaller than 0.5, 50% is a lousy summary of the confidence (ETA: common usage confidence, not frequentist confidence) that c lies between them.
If it's less than 0.5, is the confidence simply that value times 2?