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/*w*^{2}, 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 2*w.* 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 *x*^{2} 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/*w*^{2} 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). |