Normal Distribution: Understanding the Formula & Terms

AI Thread Summary
The discussion centers on understanding the Gaussian normal distribution, particularly its formula and the significance of its components. The e^-x^2 term is noted for creating the bell shape, while the expression e^-(x-xbar)^2/2sigma^2 quantifies the distance of the sample average from the mean in terms of standard deviations. The central limit theorem is highlighted as crucial for understanding how increasing sample sizes lead to convergence of sample averages to the population mean. Participants express a desire for more detailed derivations and explanations of how the normal distribution formula is formulated, particularly in relation to the binomial distribution. Overall, there is a call for clearer resources on the derivation process of the normal approximation.
O.J.
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Hello again
So we are studying the gaussian normal distribution and what I understand about it is that it helps picture many of the natural phenomena where the basica idea is probabilities are equally distributed around the average. I understand that the e^-x^2 term in the formula ensures the bell shape. But Can anyone give me some insight on how the expression was formulated and what each term means?
 
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Have you looked at the wikipedia articles for the http://en.wikipedia.org/wiki/Normal_Distribution" ?
 
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The article about the central limit theorem was helpful. From reading that and a couple of other articles about the central limit theorem I understood that increasing the sample size from a population sample makes the average of the sample or Xbar closer to the population average and drives the st. dev. to zero. I was inspecting the gaussian PDF today and what I also observed is that in the term e ^ -(x-xbar)^2/2sigma^2, the exponent is basically measuring how far the sample average is from the mean in term of the number of standard deviations squared. But, how this whole thing was put together, I would love to know. I am into little details and derivations. If anyone can help shed some light on this, please do.
 
You can derive it by taking the limit of a binomial distribution.
 
care to elaborate a bit, i really don't know where this is going? link probably?
 
Wikipedia never goes into details. I don't even know how that approximation is related to the normal distribution formula. Can you show me a link where the procedure for the derivation is clearly explained?
 
the derivation of normal approximation to binomial is in any statistics and probability textbook, so try a library if you can't find it on the internet
 

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