Normal Example

A function that uses x as the independent variable and one that uses e^x are not
the same thing.That is because e itself is a function of sorts, the product of (1 + 1/n)^n
 for a very large n. It is an irrational number (the decimal roll-out has no end, and
a transcendent number (like pi) with no geometrically stable analog ( any sufficiently
large n will generate a good chunk of it). It isparticularly useful just for this reason:
one can use it to the precision that one needs.

In the case of the normal density function, what does x mean. The expression in e
will take on different values as one moves on the x-axis, and that is the independent
variable. But taking a concrete example, as below, it is where one finds oneself on
the x-axis, and thus with respect to standard variation, that is informative.

Stanford gives the Example of the average height of women in the US. It can be
assumed to be normally distributed around a mean. The mean is 65.5 inches and
the standard deviation is at 2.5 inches plus or minus. That is to say, a majority of
women are between 63 and 68 inches.

One can create a graph with the more elaborate formula, specifying sigma at plus
or minus 2.5. And indeed, 68% of the area will show up between these values.

(Wolfram on-line definite integration calculator)

http://statweb.stanford.edu/~naras/jsm/NormalDensity/NormalDensity.html