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M O k ³ ^ ¸ Ó » v L É Ë B X ÷ J y ¶ ¸ L É < Ý é k ù G Ó ¿ x !F g b N X E l C L O ́A l C ̃f g b N X T L y A őf ꗗ A Ŗ@ Ȃǃf g b N X ڂ̐l C L O T C g ł B Ŗ@ ɔ m 点 19 N10 1 A Ŗ@ ɔ ܂ ď ŗ 8% 10% ɕς ܂ BA statement such as f(x,y) = O(g(x,y)) requires some additional explanation to make clear what is meant Still, this problem is rare in practice In addition to the big O notations, another Landau symbol is used in mathematics the little o Informally, f(x) = o(g(x)) means that f grows much slower than g and is insignificant in comparison Formally, we write f(x) = o(g(x)) (for x>) if and



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X E v j O ́A ɒB S K ܒ łɂ ܂ B ЂŎ{ H Ŕ Љ ܂ B @ ݏ i j l i C N X e L j A i j o t N G C g l i ዾ H Q j A i j C v X l iVie De France j A( L) n E W O l i R X ܖ ǁj A K ܒ ό lAnd more generally E(g(X)) for a function g = g(x) can be computed via E(g(X)) = X∞ k=−∞ g(k)p(k), discrete case, E(g(X)) = Z ∞ −∞ g(x)f(x)dx, continuous case 1 (Leting g(x) = xn yields moments for example) Finally, the variance of X is denoted by Var(X), defined by E{X − E(X)2}, and can be computed via Var(X) = E(X2)−E2(X), (2) the second moment minus the square of theĈ l l S Ⴄ g ̂Ƃ ו ܂Ŋώ@ Ȃ A X ̃I W i j \ Ɏd グ 邽 ߁A m Ȍ ʂɂ Ȃ ܂ B J } ł́A G X e e B b N ̃ b g 肾 ɁA { f B A t F C X g g g ܂ŁA ʂɔ l X ȏ ̃f b g ̕ 邽 ߂̃T r X p ӂ Ă ܂ I



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The statement "f(x) is O(g(x))" as defined above is usually written as f(x) = O(g(x)) Some consider this to be an abuse of notation, since the use of the equals sign could be misleading as it suggests a symmetry that this statement does not have As de Bruijn says, O(x) = O(x 2) is true but O(x 2) = O(x) is not Knuth describes such statements as "oneway equalities", since if the sides couldE F f B O h X B ^ h X ̓X h X M ł̓E G f B O h X p E Q X g p J h X A t H } X c A r A q A A N Z T E ܂ŁA T C Y A ނƂ L x Ɏ 葵 Ă ܂ BL s t ̃G X e T ł B t w k R ƃA N Z X ɕ֗ B g A _ C G b g Ȃǂ̃ j Ă A n C p i C t ̓ A x A } I C E V F ґ Ɏg p { p ܂ B S g ̌ s 悭 f g b N X łȏz ɃA v ` B u ߁v u ق v ̃_ u ʂŃX s f B ɃT C Y _ E B



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