By Frank A. Haight (Eds.)

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**Example text**

TYPEI11 DISTRIBUTION Although this function has no simple form, it occurs often enough in applied mathematics to deserve a special name and symbol. n, x) + r ( n ,x) =qn). (9) It is possible to express the cumulative Poisson distribution in terms of r ( n , x ) : A proof of this rather surprising fact may be obtained by repeated integration by parts of r ( n 1, A) for integral n. + TYPEI DISTRIBUTION. 7 MOMENTS AND EXPECTATION 19 This cumulative distribution defines the incomplete beta function, as follows: Because of their close connection with the gamma and beta functions respectively, the Type I11 distribution is also called the gamma distribution, and the Type I distribution is also called the beta distribution.

5 we have given only a numerical example of this important type of distribution, but in Chapter 4 we will amplify and make use of the concept. Similarly, the brief introduction to multivariate distributions will be enlarged in Chapter 7, where such distributions are used. The student unfamiliar with transformations and generating functions may well wish for a more systematic classification of the auxiliary functions. The fact is that these transformations arose in different parts of applied mathematics for certain specific purposes, and that as a consequence, the information available, though vast, is not orderly.

EXAMPLE 3. x individual. = age of an individual, y = weight of same EXAMPLE 4. n = number shown on top in a throw of a die, N = number shown on nearest side. EXAMPLE 5 . x = space between consecutive cars, n = number of passengers in leading car. EXAMPLE 6. x = space between consecutive cars, y = relative speeds of two cars. The domains of definition above, and the corresponding probability distributions, are bivariate. Examples 3 and 6 are continuous both ways; Example 4 is discrete both ways; Examples 1, 2, and 5 are discrete one way and continuous the other.