Saeed,

I believe you are attempting to define a confidence interval for the mean mu (aka expected value). This is close to what you wrote, but should really be

[Xbar - Z*sigma/sqrt(n), Xbar + Z*sigma/sqrt(n)],

where Xbar = (1/n)*SUM_{i=1:n} Xi and Z is a critical value obtained from a known probability distribution and a specified confidence level.

If n is large enough, the central limit theorem allows you to use the standard normal distribution for the probability distribution. The confidence level can be pretty much anything you like, but it is often taken as alpha = .05 (i.e. 95% confidence). Note that use of the above confidence interval for mu requires that the standard deviation (sigma) is known.

Xbar is the "sample mean" or "average" if you like. It is an estimate of mu that gets better (i.e. closer to mu) as the sample size n increases. Think of mu as a fixed quantity that is independent of n. It makes sense that the length of the above interval decreases as n increases because Xbar gets closer to mu as n increases.

Xbar is a random variable because it it defined in terms of the independent and identically distributed random variables X1, X2, ..., Xn. In other words, Xbar changes each time a random sample is collected. The standard deviation of Xbar is sigma/sqrt(n). This says that the variance of Xbar becomes smaller and smaller as n increases, which again makes sense for the same reason.

Since the above interval involves Xbar, it constitutes a random interval. When you evaluate Xbar for a particular random sample instance, you can evaluate this interval as well. The result is one instance of a random interval. We call that instance a confidence interval.

There is nothing random about this interval once you evaluate it. It will either contain the constant mu or it won't. It makes no sense to ask "what is the probability that this particular instance of the random interval contains mu?". It does however make sense to ask "what is the probability that the next (not yet evaluated) instance of the random interval contains mu?".

The idea of a confidence interval centers around the idea of repeated experiments, where an experiment consists of collecting instances of X1, X2, ..., Xn and using them to evaluate instances of [Xbar - Z*sigma/sqrt(n), Xbar + Z*sigma/sqrt(n)]. As you repeat this experiment many times, the percentage of intervals that contain mu will approach 100*(1-alpha)% (e.g. 95% for alpha = 0.5).

The ideas behind confidence intervals are somewhat subtle and may take a little time to sink in. Best wishes.

The confidence interval refers to (the unknown) mu.Having larger sample sizes means having more information about the unknown what translates into narrower confidence intervals (on average).

What you want is probably a prediction interval (https://en.wikipedia.org/wiki/Prediction_interval). This is about where one may expext realizations of the random variable (X). These will be somewhere around mu (which is estimated just as for the confidence interval), but acknowledges that the variance (or sigma) will always add to the uncertainty, no matter how well mu is estimated.

To my understanding, your probability distribution function does not behaves generally/normally, but it works well probably for very low or very large values of concentrated parameters, i.e., uses ad hoc methods for such solutions. In order to obtain a general solution, i.e., model validation for the whole domain of interval, I would like to suggest you saddle point method of approximation or Laplace approximation for high dimensional integrals. At the end you will need Bayesian maximum likelihood or expectation maximization or variation methods.

Dear Gregory

Thank you for your thorough explanations on the definition of Confidence Interval.