
Advanced Statistics Advanced Probability and Statistics Math Forum 
 LinkBack  Thread Tools  Display Modes 
January 20th, 2017, 06:19 AM  #1 
Newbie Joined: Feb 2012 Posts: 3 Thanks: 0  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? 
January 20th, 2017, 02:09 PM  #2 
Senior Member Joined: Sep 2015 From: CA Posts: 1,107 Thanks: 578 
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. 
January 20th, 2017, 06:15 PM  #3  
Global Moderator Joined: May 2007 Posts: 6,214 Thanks: 491  Quote:
 
January 21st, 2017, 07:57 AM  #4 
Newbie Joined: Feb 2012 Posts: 3 Thanks: 0 
@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... 
January 21st, 2017, 04:55 PM  #5  
Global Moderator Joined: May 2007 Posts: 6,214 Thanks: 491  Quote:
 

Tags 
samples, sigma 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Convenience Samples  EvanJ  Probability and Statistics  1  September 4th, 2016 05:50 AM 
What are the number of rainfalls in each samples?  Jessicasamide  Algebra  1  November 24th, 2012 06:47 PM 
Samples and Populations  jhon13  Advanced Statistics  1  May 23rd, 2012 11:40 PM 
how to find : sigma f/sigma x  luketapis  Calculus  2  March 27th, 2012 11:30 AM 
Samples vs Experiments  hunnybee  Algebra  1  June 23rd, 2011 05:26 AM 