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EaX b = aEX b This says that expectation is a linear operator Variance The variance of a random variable tells us something about the spread of the possible values of the variable For a discrete random variable X, the variance of X is written as Var(X) Var(X) = E (X – m) 2 where m is the expected value E(X). Æ Ç È É Ê Ë Ë Ê Ì È Í Æ Î Ï Ð Ñ Ñ Ò Ó Ô Õ Ö × Ø Ù Ú Û × Ö Ü Ý Þ ß Ù à Þ á Ö â ã à á Ö ä å æ á × Ö ç à è â é Ø á ê â ë ì í í × á å è ×Ø î Ù à êá Ö Ú ç Ø ï Ù ð Ý à Ø ì Ü á ê ñ ï × Ö á ò á æ á Ö Û Ý à Ù ó à á ô × á õ ì à ö Ù â ÷ ø ù. Ø ¿ n o C I } X É æ é y b g R ¿ Ì ¶ Y m R ` § E ù ø ¬ n î ñ û W ¯ ¼ F k _ ­ Ç ß ª v E î ñ Z ^ m æ g å Ì n ¼ Ì i j n ï d C y Ø.

F5 28 i y j17 F00 J \ ꏊ F Z { ؃X p f b N X ꗿ F 00 h N ` P b g. Proof lnexy = xy = lnex lney = ln(ex ·ey) Since lnx is onetoone, then exy = ex ·ey 1 = e0 = ex(−x) = ex ·e−x ⇒ e−x = 1 ex ex−y = ex(−y) = ex ·e−y = ex · 1 ey ex ey • For r = m ∈ N, emx = e z }m { x···x = z }m { ex ···ex = (ex)m • For r = 1 n, n ∈ N and n 6= 0, ex = e n n x = e 1 nx n ⇒ e n x = (ex) 1 • For r rational, let r = m n, m, n ∈ N. Recall that if B = {b1,,b n} is a basis for V and v = P x ib i then we write vB for the column vector vB = x1 x n Definition 12 If f is a bilinear form on V and B = {b1,,b n} is a basis for V then we define the matrix of f with respect to B by.

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# $ % & ’ ( $ % ) * , % / % 0!"1 23 45 67 ;. Equations Tiger Algebra gives you not only the answers, but also the complete step by step method for solving your equations g(x)=1/2f(x) so that you understand better. Proof lnexy = xy = lnex lney = ln(ex ·ey) Since lnx is onetoone, then exy = ex ·ey 1 = e0 = ex(−x) = ex ·e−x ⇒ e−x = 1 ex ex−y = ex(−y) = ex ·e−y = ex · 1 ey ex ey • For r = m ∈ N, emx = e z }m { x···x = z }m { ex ···ex = (ex)m • For r = 1 n, n ∈ N and n 6= 0, ex = e n n x = e 1 nx n ⇒ e n x = (ex) 1 • For r rational, let r = m n, m, n ∈ N.

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