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January 20th, 2017, 05:19 AM   #1
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All N samples within one sigma

If I have a random variable which has a normal distribution with mu and sigma, what is the probability that if I draw N samples that they are all less than one sigma away from each other?

I am not having any luck coming up with how to calculate this answer. I did write a program to try to determine the answer empirically.

10,000,000 tests per draw set:
Draw: 2, All with in sigma: 84.27%
Draw: 3, All with in sigma: 66.63%
Draw: 4, All with in sigma: 50.96%
Draw: 5, All with in sigma: 38.17%
Draw: 6, All with in sigma: 28.16%
Draw: 7, All with in sigma: 20.57%
Draw: 8, All with in sigma: 14.90%
Draw: 9, All with in sigma: 10.71%
Draw: 10, All with in sigma: 7.69%

Would it be something like:
$\displaystyle \int_{x-\sigma}^{x+\sigma} \frac {1} {\sigma \sqrt{2 \pi}} e^{\frac {-(x{'}-\mu)^2}{(2 \sigma^2)}} dx{'}$

Any thoughts?
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January 20th, 2017, 01:09 PM   #2
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I believe you'll want to use order statistics.

All of the samples being within a sigma of each other implies

$\max(X_i) - \min(X_i) \leq \sigma$

So you're going to have to find the joint distribution of the min and max order statistics and then integrate that over the area that corresponds to the expression above.

This isn't impossible but it's fairly involved. Give it a shot and come back w/specific questions if you run into problems.
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January 20th, 2017, 05:15 PM   #3
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Quote:
Originally Posted by histrung View Post
If I have a random variable which has a normal distribution with mu and sigma, what is the probability that if I draw N samples that they are all less than one sigma away from each other?

I am not having any luck coming up with how to calculate this answer. I did write a program to try to determine the answer empirically.

10,000,000 tests per draw set:
Draw: 2, All with in sigma: 84.27%
Draw: 3, All with in sigma: 66.63%
Draw: 4, All with in sigma: 50.96%
Draw: 5, All with in sigma: 38.17%
Draw: 6, All with in sigma: 28.16%
Draw: 7, All with in sigma: 20.57%
Draw: 8, All with in sigma: 14.90%
Draw: 9, All with in sigma: 10.71%
Draw: 10, All with in sigma: 7.69%

Would it be something like:
$\displaystyle \int_{x-\sigma}^{x+\sigma} \frac {1} {\sigma \sqrt{2 \pi}} e^{\frac {-(x{'}-\mu)^2}{(2 \sigma^2)}} dx{'}$

Any thoughts?
the integration limits are $\displaystyle \mu -\sigma ,\mu +\sigma $
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January 21st, 2017, 06:57 AM   #4
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@mathman:
I think that would be the probability that the random variable drawn is with in 1 $\displaystyle \sigma$. I need to know if multiple sample are drawn (N), the probability they are all within 1 $\displaystyle \sigma$ of each other.

@romsek:
Googling, reading and digesting...
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January 21st, 2017, 03:55 PM   #5
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Quote:
Originally Posted by histrung View Post
@mathman:
I think that would be the probability that the random variable drawn is with in 1 $\displaystyle \sigma$. I need to know if multiple sample are drawn (N), the probability they are all within 1 $\displaystyle \sigma$ of each other.

@romsek:
Googling, reading and digesting...
romsek answered
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