Normal (Gauss) Distribution questions

[Master J]

Am I correct in my understanding of the standard normal distribution:

An idealized probability distribution which can be used to approximate many distributions which arise in practical applications.
(WHat are some of these applications?)

How exactly does the standardized normal distribution related to the Normal (Gauss) distribution, as in, why is it needed? Why are the results obtained from the standardized?

As a physics student, a lot of the details of these would be beyond my scope, but I would like to have an idea so to better understand it.

Hope someone can help clear thse up! Thanks!

 

[John Creighto]

Am I correct in my understanding of the standard normal distribution:

An idealized probability distribution which can be used to approximate many distributions which arise in practical applications.
(WHat are some of these applications?)

I think if the standard deviation is small relative to the mean then it is an okay for a lot of applications. For large standard deviations it won't always be a good choice given that the tails go to infinity and the higher order statistics are determined completely by the mean and standard deviation.

How exactly does the standardized normal distribution related to the Normal (Gauss) distribution, as in, why is it needed? Why are the results obtained from the standardized?

They are related by a linear change of variables.

 

[quadraphonics]

One of the main justifications for employing a Gaussian model is given by the Central Limit Theorem. This theorem states that suitable linear combinations of suitably-behaved random variables will, asymptotically, display a Gaussian distribution, regardless of the distributions of the individual random variables being combined.

So, any time you are looking at a random variable that is produced by linearly combining lots of well-behaved random variables (which is common in physics and engineering), then you can justify assuming that the result is Gaussian.

Another justification is that the binomial distribution can, in the limit, be nicely approximated by a Gaussian distribution, so there is also a connection to discrete random variables.

 

[rbeale98]

How exactly does the standardized normal distribution related to the Normal (Gauss) distribution, as in, why is it needed? Why are the results obtained from the standardized?

The standard normal distribution is helpful in engineering practice because it allows textbooks to print a table giving the area under the distribution curve between 0 and a given z value. You can calculate the z value for any normal distribution using the mean and stdev. Z value is interpreted as "the number of standard deviations away from the mean" and can be positive or negative.

For example you can look up the 99 percentile Z value, and then calculate the 99th percentile threshold in your application if you know your mean and stdev.