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E D L H c ab , F a H a A a L J , D Sa R a c , NIOSH. NDOH 0DUNHW O ;. 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.

E(X)= X x X xpX (x) = X6 i =1 i 1 6 = 6 = 21 6 =3 1 2 (12) 6 RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS 173 Second example computing an expected value Consider a group of 12 television sets, two of which have white cords and ten which have black cords Supposethree of them are chosen at ran. = $ Á ^ Ä Ì < $ × Ø Ù Ú À Û Õ Ö Ã 5 % Ú Â Û 5 %?. Uploaded By shihanlin0105 Pages 84 This preview shows page 48 63 out of 84 pages.

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Ú ¹ E Ú ¹ Ö Ã Ë µ, À Æ ó Ë ¢ « £ } R e ~ R å Ú e w E x e ¯ ` b { } \ Ë w Ú ¹ x z x å Ú e t R å Ú e t m M o x z å Ú § ÷ ² ÿ w å Ú § ÷ Ì t S Z å Ú j ¿ q å Ú § ÷ ÿ w y Ú j Ú ¹ ù ` h w z R e t m M o x y Ú j Ú ¹ ¯ ` b {. Má ¦ »*Ë>& Ù0¯*Ë (>' b À7 _ X 8 Z $ª ¦ N4 4 %31¤4 %31¤'ö#1" >/>, À7 *Ë' l g À7 Ç X Má ¦ »*Ë>& Ù0¯*Ë (>'>/ ¡. X>0 Note From the pdf of the gamma distribution, if we set = 1 and x= 1 we get f(x) = e We see that the exponential distribution is a special case of the gamma distribution { Find cdf of the exponential distribution { Find the mean of the exponential distribution { Find the variance of the exponential distribution.