## An Introduction to Probability Theory by P. A. Moran

By P. A. Moran

Книга An creation to likelihood concept An advent to chance concept Книги Математика Автор: P. A. Moran Год издания: 1984 Формат: pdf Издат.:Oxford collage Press, united states Страниц: 550 Размер: 21,2 ISBN: 0198532423 Язык: Английский0 (голосов: zero) Оценка:"This vintage textual content and reference introduces likelihood conception for either complicated undergraduate scholars of information and scientists in similar fields, drawing on actual purposes within the actual and organic sciences. "The ebook makes likelihood exciting." --Journal of the yankee Statistical organization

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Example text

2 Show that if X’s are independent and have the same continuous distribution function F then for any n = 1, 2,… record indicators ξ1, ξ2,…, ξn and maximal value M(n) are independent random variables. Now let N(n) denote the number of the strong upper record values among the random variables X1, X2,…,Xn. 3 Record Indicators 49 N ðnÞ ¼ n1 þ n2 þ Á Á Á þ nn ; n ¼ 1; 2; . : ð3:3:3Þ Any information about the distribution of N(n) helps us to investigate the distribution properties of record times L(n).

It means that it is enough to study in details only one of these record types. Below we will deal as a rule with the upper record times and the upper record values. Denote X ð1Þ\ X ð2Þ\ Á Á Á the corresponding record values (strong upper record values) in the sequence X1, X2,··· and let 1 ¼ Lð1Þ\Lð2Þ\ Á Á Á be the corresponding record times. Introduce also record indicators nn ; n ¼ 1; 2; . ; which take values 0 and 1, and mark the appearance of record values, that is nn ¼ 1, if Xn [ M ðn À 1Þ; and nn ¼ 0; otherwise.

Mðn À 1Þ ! mðnÞ ! Á Á Á. The appearance of the sign of strong inequality in this sequence signiﬁes the appearance of the strong lower record. Analogously, if Xn ¼ mðn À 1Þ for some n, one deals with a weak lower record value. Note that in all options X1 is taken as the ﬁrst (upper or lower, strong or weak) record value. © Atlantis Press and the author(s) 2015 M. B. 2991/978-94-6239-136-9_3 45 46 3 Record Times We mentioned above that these two constructions (upper and lower records) are symmetrical in some sense.