Definition 4

  Any function which is the negative of the logarithm of a frequency distribution will be called a W-function. If the frequency distribution is proper, the corresponding W-function will also be called proper. From the definition of a proper frequency distribution it follows that if w(x) is a proper W-function, then

(a)

w(x) is monotonic decreasing

(b)

$\infty \geq w(x) \geq 0$

(c)

$\lim_{x\to-\infty} w(x) = \infty$

(d)

$\lim_{x\to\infty} w(x) = 0$

(e)

there is at least one value of x such that $\infty \gt w(x) \gt 0$