I am aware that for large sample sizes the sample mean tends towards a normal distribution but this is for either small sample sizes or for a more exact distribution than an approximate normal distribution. I think my idea works but part of the formulation of my conclusion is weak so I am posting it here for some feedback.

Starting with the simple case of a sample size of 2.

Suppose we are sampling from a continuous distribution with density function f(x) and range

. We take 2 samples which are random variables X

_{1} and X

_{2}. The sample mean is

Given that the first sample takes the value a the second sample must take the value

in order for the sample mean to equal

The probability that X

_{1} takes the value a and X

_{2} takes the value b is f(a)f(b)

This is the probability for the case where x

_{1} took the value a, to account for all values of x1 the expression is integrated over the entire range.

After evaluating the integral the resulting expression is the distribution of the sample mean.

Now for the case where the sample size is 3

The sample mean is

Given that the first sample takes the value a and the second sample takes the value b the third sample must take the value

in order for the sample mean to equal

The probability that X1 takes the value a, X

_{2} takes the value b and X3 takes the value c is f(a)f(b)f(c).

This is the probability for the case where x

_{1} took the value a, and x

_{2} took the value c, to account for all values of x

_{2} and x

_{2} the expression is integrated over the entire range.

After evaluating the integral the resulting expression is the distribution of the sample mean.

You can see how this theory extends for a sample size n.

Could I please have some feedback about this idea?