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題名 | 獨立伯努利變數和之變異數與香儂熵=On the Variance of Sum of Independent Bernoulli Random Variables and Shannon Entropy |
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作者 | 繆紹昌; 陳志賢; 劉家頤; Miao, Shao-chang; Chen, Chih-sheng; Liu, Chia-yee; |
期刊 | 修平學報 |
出版日期 | 20110300 |
卷期 | 22 2011.03[民100.03] |
頁次 | 頁35-43 |
分類號 | 319.5 |
語文 | chi |
關鍵詞 | 蓋理論; 蕭爾-凸性質; 香儂熵; 凸多邊形區域; Majorization; Schur-convexity; Shannon entropy; Convex polygonal region; |
中文摘要 | 假設某國中某班共有n名學生,令pi(0≦pi≦1)為第i位學生能順利進入理想高中之機率。令Xi 為參數pi之伯努利隨機變數,則S=X1+X2+...+Xn為能進入理想高中之總人數。在p1+p2+...+pn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 pi(0≦pi≦1) reflecting the student’s probability of being admitted to an ideal high school. Let Xi(1≦i≦n ) be independent Bernoulli random variables with parameters pi. Then S=X1+X2+...+Xn is the number of the students in the class who will be admitted to an ideal high school. Assuming that p1+p2+...+pn 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. |
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