A normally distributed random number distribution is just the famous "Bell curve". There are many ways to describe it, but we will use the form:

An important property of a probability distribution is for it to be normalized, that is, that the sum of all possible outcomes be exactly 1. Thus, we require:

This kind of integral is often treated in texts on statistical mechanics or quantum mechanics. For example, in [2] we find:

.

Making the substitution a = 1/w², we find:

and thus,

.

So f(x) is a properly normalized distribution.

The Bell curve is a peaked function, so it is useful to know the height and width of the curve. The curve is obviously symmetric about x = 0, so the peak height is given by just evaluating the function at x = 0. So,

.

Since the function involves an exponential, we adopt the "full width at 1/e max" (FW1/eM) as a measure of the width of the function. Since the function drops to 1/e of its peak value when , the FW1/eM is 2w. Note that the height and width of the function are closely connected and are inversely proportional to one another.

Another useful quantity is a measure of the average size of each x. But since the distribution is symmetric around zero, the mean value of x is zero. Instead, we calculate the mean value of x² and then take the square root giving the root mean square, x_rms.

Again, integrals like these are treated in textbooks on statistical mechanics and quantum mechanics. Ref. [2] gives:

.

Using the substitution a = 1/w² as before, we obtain:

Hence,

.

Note the simple relationship between x_rms and the FW1/eM:

[2] D. V. Schroeder, "An Introduction to Thermal Physics" (Addison Wesley, Boston 2000).