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 alt="images"/> elements by Theorem 3.2.3, of which images are of the form images. The number of pairs with elements distinct is images.

      We can generalize these considerations as follows:

      Definition 3.2.1 An ordered sequence of images elements selected without replacement from a set of images distinct elements images is called a permutation of images out of images elements.

equation

      Proof: The argument here repeatedly uses the “operation” principle: the first choice can be made in images ways, the second in images ways, the imagesth in images ways.

      Corollary 3.2.6 The set of images elements can be ordered in

       distinct ways.

      The product (3.2) occurs often and has a special symbol:

equation

      to be read “images factorial.” We have therefore

      For a reason that will become apparent later, we adopt the convention

      (3.4)equation

      Example 3.4

      The letters I, I, I, I, M, P, P, S, S, S, S are arranged at random. What is the probability that the arrangement will spell MISSISSIPPI?

      Solution

      We can solve this problem treating the choices of consecutive letters as “operations.” The first operation must give the letter M; hence, there is only one way of choosing it. The next letter (out of the remaining 10) must be an I, and it can be selected in four ways. Proceeding in this way, the sequence of consecutive 11 choices leading to the word MISSISSIPPI can be performed in images ways, which equals images. On the other hand, the total number of ways one can perform the operations of consecutively choosing letters from the set is images. Consequently, the required probability equals

      In this solution, the letters are regarded as distinguishable, as if we had four letters images, labeled images and images, and similarly for the other letters. In this case, the numerator and denominator are, respectively, the number of ways one can order the set of distinguishable letters so as to form the word MISSISSIPPI and the total number of orderings. Alternatively, one can regard the identical letters as indistinguishable, and in this case, we have only one way of ordering them so as to spell the required word, and a total of images distinguishable ways of ordering these letters. Indeed, the denominator here represents the number of ways of permuting letters so as to leave the arrangement invariant. Now,

equation

      Example 3.5 Birthday Problem

      The following problem has a long tradition and appears in many textbooks. If images randomly chosen persons attend a party, what is the probability images that none of them will have a birthday on the same day?

      Solution