Definition 2 (Gnedenko and Kolmogorov (1954), p. 40)

  The frequency distributions F₁(x) and F₂(x) are said to belong to the same type if for some constants a > 0 and b the following equation holds:

F₂(x) = F₁ (ax+b)

or, what is the same,

$$F_1 (x) = F_2 \left( \frac{x}a - \frac{b}a \right).$$

Since the property of belonging to the same type is symmetrical and transitive, the totality of distribution functions fall into mutually disjoint types.

It is easy to see that all normal laws of distribution form one type, the normal type; all improper distribution functions form the improper type. The types of distribution functions other than the improper type are called proper.