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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, xrms.

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 xrms and the FW1/eM:

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