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Fisher's inequality, is a necessary condition for the existence of a balanced incomplete block design which satisfies certain prescribed conditions in combinatorial mathematics. Outlined by Ronald Fisher, a population geneticist and statistician, it concerns with the design of experiments studying the differences among several different varieties of plants, under each of a number of different growing conditions, called "blocks". Let: * ''v'' be the number of varieties of plants; * ''b'' be the number of blocks. It was required that: * ''k'' different varieties are in each block, ''k'' < ''v''; no variety occurs twice in any one block; * any two varieties occur together in exactly λ blocks; * each variety occurs in exactly ''r'' blocks. Fisher's inequality states simply that :: == Proof == Let the incidence matrix M be a ''v×b'' matrix defined so that M''i'',''j'' is 1 if element ''i'' is in block ''j'' and 0 otherwise. Then B=MMT is a ''v×v'' matrix such that B''i'',''i'' = ''r'' and B''i'',''j'' = λ for ''i'' ≠ ''j''. Since ''r'' ≠ λ, det(B) ≠ 0, so rank(B) = ''v''; on the other hand, rank(B) = rank(M) ≤ ''b'', so ''v'' ≤ ''b''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Fisher's inequality」の詳細全文を読む スポンサード リンク
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