题名

獨立伯努利變數和之變異數與香儂熵

并列篇名

On the Variance of Sum of Independent Bernoulli Random Variables and Shannon Entropy

DOI

10.29548/BGYY.201103.0003

作者

繆紹昌(Shao-Chang Miao)；陳志賢(Chih-Sheng Chen)；劉家頤(Chia-Yee Liu)

关键词

蓋理論 ； 蕭爾－凸性質 ； 香儂熵 ； 凸多邊形區 ； Majorization ； Schur-Convexity ； Shannon Entropy ； Convex polygonal Region

期刊名称

修平學報

卷期/出版年月

22期（2011 / 03 / 01）

页次

35 - 43

内容语文

繁體中文
中文摘要

假設某國中某班共有n名學生，令p(下標 i)(0≤p(下標 i≤1)為第i位學生能順利進入理想高中之機率。令X(下標 i)為參數p(下標 i)之伯努利隨機變數，則S=X1+X2…+X(下標 n)為能進入理想高中之總人數。在p1+p2+…p(下標 n)為一固定常數的限制下，以兩種方法找出Var[S]之極大值與極小值之條件，也建立出Var[S]之極值與香儂熵之關係。
英文摘要

There are n students in a class. The ith student is evaluated and assigned a constant P(subscript i) (0≤P(subscript i)≤1) reflecting the student's probability of being admitted to an ideal high school. Let X(subscript i) (1≤i≤n) be independent Bernoulli random variables with parameters P(subscript i). Then S=X1+X2+…X(subscript n) is the number of the students in the class who will be admitted to an ideal high school. Assuming that P1+P2+…P(subscript n) is a fixed constant, the maximum and minimum values of Var[S] are obtained using two different methods. The notions of majorization and Shannon entropy relevant the problem are defined and discussed. The relationships between the extremal values of Var [S] and Shannon entropy are also established.
主题分类 人文學 > 人文學綜合
基礎與應用科學 > 基礎與應用科學綜合
工程學 > 工程學綜合
工程學 > 機械工程
社會科學 > 社會科學綜合
参考文献
  1. Bose, R.(2003).Information Theory, Coding and Cryptography.Singapore:McGraw Hill.
  2. Boyd, S.,Vandenberghe, L.(2004).Convex Optimization.New York:Cambridge University Press.
  3. Gass, S.(1994).Linear Programming.Singapore:McGraw Hill.
  4. Hardy, G.,Littlewood, J. E.,Polya, G.(1952).Inequalities.New York:Cambridge University Press.
  5. Marshall, A. W.,Olkin, I.(1979).Inequalities: Theory of Majorization and Its Applications.New York:Academic Press.
  6. Rao, M.,Chen, Y.,Vemuri, B. C.(2004).Cumulative Residual Entropy A New Measure of Information.IEEE Trans. on Information Theory,50(6),1220-1228.
  7. Ross, S.(2006).A First Course in Probability.Singapore:Pearson Education International.
  8. Shannon C. E. (1948 ) , A Mathematical Theory of Communication, Vol 27, pp 379-423, 623-656, The Bell System Technical Journal. [http://cm.bell-labs.com/cm/ms/what/shannonday/shannon1948.pdf]
  9. Valentine, F. A.(1968).Convex Sets.New York:McGraw Hill.
  10. 李天岩(1989)。熵（Entropy）。數學傳播，13(3)
  11. 楊重駿,楊照崑(2002)。蓋理論（Theory of Majorization）及其在不等式上的應用。數學傳播，6(4)
print cancel