Probability and Statistical Inference. Robert Bartoszynski
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Название: Probability and Statistical Inference

Автор: Robert Bartoszynski

Издательство: John Wiley & Sons Limited

Жанр: Математика

Серия:

isbn: 9781119243823

isbn:

СКАЧАТЬ to partitioning the set of size images into two subsets, one of size images and the other of size images. The number of such partitions is, by definition,

equation

      The theorem below generalizes this scheme.

      Theorem 3.4.1 Let images be positive integers such that images. The number of ways a set of images elements can be partitioned into images subsets of sizes images equals

      Proof: A partition above can be accomplished in steps: First, we choose images out of images elements to form the first subset of the partition. Next, we choose images elements out of the remaining images elements, and so on, until we have images elements, from which we choose images to form the next‐to‐last subset. The remaining images elements form the last subset This can be accomplished, in view of Theorem 3.2.2, in

equation

      As a generalization of Newton's binomial formula, we have

      Theorem 3.4.2 For every integer images,

      where the summation is extended over all subsets images of nonnegative integers with images.

      Proof: In the product images, one term is taken from each factor so that the general term of the sum has the form images with images. From Theorem 3.4.1, it follows that the number of times the product images appears equals (3.26).

      The theorem is illustrated by the following example:

      Example 3.14

      Suppose that one needs the value of the coefficient of images in the expression images. One could argue that in the multiplication images there are 10 factors, and each term will contain one component from each set of parentheses. Thus, choosing images from 2 out of 10 pairs of parentheses, images from 3 out of 10, and so on, amounts to partitioning 10 pairs of parentheses into four classes, with sizes images and images. The total number of ways such a partition can be accomplished is the coefficient of images, and equals images.

      An approximation to images is given by the so‐called Stirling's formula.

      Theorem 3.4.3 (Stirling's Formula) We have

      where the sign images means that the ratio of the two sides tends to 1 as images.

      We shall not give the proof here, but interested readers can find it in more advanced texts, for example, in Chow and Teicher (1997).

      Example 3.15

      A СКАЧАТЬ