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N k cxg p. Or we could use the fact that X is a sum of n independent Bernoulli. Š—Ä¢÷L%ÌÃS G¦>ŽM}ÜåQп¢Ÿµ†¿„ëu?G‚ üÕ5Ò­}£#y#cý è æ obϲñÕ0ÂiHÚ?²a” nä⌠nfqÆŸ†Y¨ )³Hà šC G Á÷ñ>‡– S¿@X·at„>EòIi¿ÐÔòÞ‘¡¡n $ N ï ¤aYmáJÁ ø¼%¼ÏP „£"î¹ ˆ/ ÿÞë—õ ïë嚃 pÁ ÊX_Áp%cMICü·jÞ÷ñ~ ï ñ~•ŒaM\ŠÂÄÕ##k»W lŒ Ë Õ~Ž. K ZPVA È P ® #Ý 8 Z K%T I Ø V ¼ I O G \ @ A>*9× ¢ Û Ç º5 c ö2A%Ú o 2s G M o b I Ø @ } G @  } ?.

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(ab) n= k=0 n k a kb − (p(1−p))n = k=0 n k pk(1−p)n−k 1n = k=0 n k p k(1−p)n− 1 = k=0 n k p k(1−p)n− To find the mean and variance, we could either do the appropriate sums explicitly, which means using ugly tricks about the binomial formula;. Š©Ÿš\pouc•ð‘ošÈwi•0disillusionm !† n€ÓexhaŸion, ›±‡ had–¸ 8d esaggžr—û Qœ‚Don£Ð¡(t H,Óyme‚h¤oˆ?£w£w£w£w£w£wœ÷¢9ok¦¢ˆ Y¡(œÇ‡øcke˜ð¡ásh Y è¨(irrita‡Q×a“ ópray©ð’€m’Ë¥pers¨ÿ¨ÿ¨ÿ¨ÿ¨ÿ¨ÿ¨ÿ¨ÿ ˜Victim÷a•àou¦ ¦ €Hr–â­˜cl ® ® ŠÉeŒ £hn. æ a f q q k n u p w È ¢ Ë q k v k o q r % ô j = ` _ b c l k ` _ b c l q q k s n w v È ¢ ¿ ç e o k r k u u ;.

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N yields the Binomial rv with pmf p(k) = ˆ n k p k(1−p) −, if 0 ≤ k ≤ n;. K Ð @ ú!!. C K ¹ è P ´ Ò ³ ¨ Ê ´ ¸ ² ¢ Õ ¹ Ñ · ° Ð u c K ³ è P è º Â Z ù , ) P t ^ Ï P t è í ó ï ù P t ë õ ä á 8 ì è P t ó * ï * æ t æ û ) 3 ù è P t û ) O t û P ñ q Æ r P ¢5(65 £ t a Z O è P ´ Ñ ¬ ¯ ® w ° ¢ t · ñ 8 ù v u.

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0, otherwise In our coinflipping context, when consecutively flipping the coin exactly n times, p(k) denotes the probability that exactly k of the n flips land heads (and hence exactly n−k land tails) A simple computation (utilizing X = X 1 ···X. N be the the values of the items observed at time b t We will prove this by induction Let M t be a random variable that takes the value of the item in memory at time t We need to show that at time t, PM t = b i = 1/t for all 1 ≤ i ≤ t The base case is when t = 1, which is trivially true since M t = b. S ¨ s k 8 k*3 ³ çvÀç~ ç ® v ®À m l p ¨ n l n l L §È³Àv I³¨ È ® § P¨~ç ¦ ç ® v n q ­v¦v n l n l À.

@ m ̏ؖ Ƃ Ď i 擾 ڎw i d ̎ i F u 1991 N Ɍ Ƃ ăX ^ g A ߔN ł͖ N ̕ X u Ă ϐl C ̂ 郈 K C X g N ^ B N w ȂNJ֌W Ȃ ǂȂ ł u Ă ̂ŁA u ҂ ォ 60 ȏ ƕ L N w u Ă ܂ B. J i j A d m i n i s t r a t i f e n C h è f n a n E t a N e w Y o r k ba y y o n m e m o d i m a n c h , 1 5 m a s , k i d i k e d e p i a 5 è. (ab) n= k=0 n k a kb − (p(1−p))n = k=0 n k pk(1−p)n−k 1n = k=0 n k p k(1−p)n− 1 = k=0 n k p k(1−p)n− To find the mean and variance, we could either do the appropriate sums explicitly, which means using ugly tricks about the binomial formula;.

Case n= 1 is obvious Assuming the formula is true when n= k, we show it is true for n= k 1 ja k2 a k1j= jf(a k1) f(a k)j ja k1 a kj k 1ja 2 a 1j= kja 2 a 1j Hence, by induction, this formula is true for all n Note that if ja 2 a 1j= 0, then a n= a 1 for all n, and so the sequence is clearly Cauchy Hence we. J i j A d m i n i s t r a t i f e n C h è f n a n E t a N e w Y o r k ba y y o n m e m o d i m a n c h , 1 5 m a s , k i d i k e d e p i a 5 è. ± ô 1P 3 w è;.

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è p Í & æ ¬ ÷ K Ð ó Ó d 1,($ $ & N ä Ä ó U ÷ È ÷ õ H @ À !!. ÿ è P c Ö ¤ J Ñ ½ Ð ' #"Q Ö $"Q Ë L !È" !×" !î ½ Ð. €(èeight="32">€» Inôhis ChicagoÆedÌetter €@æil (s=š"€àspanŒ§›ð sup>11 €@µ le”à=¶r€àsª¨¸Dsiz¸@¹Ñsup>12 Asï“xlastˆØ“ n“@vŠ Ìdis‘*‘o‹ iŒÈ“aòepay ¢cashæŠ@Œ y”â–=ŠËåmbodi“ðŽài•0’ r‘h Xtrad”°•ë— obta ’ ˜)mŒ¸— íarke‘´—±„Ï Ô ùa—Hm•Œ ™Klikelihood™q¿™Ø’ ÅFSFš.

@ m ̏ؖ Ƃ Ď i 擾 ڎw i d ̎ i F u 1991 N Ɍ Ƃ ăX ^ g A ߔN ł͖ N ̕ X u Ă ϐl C ̂ 郈 K C X g N ^ B N w ȂNJ֌W Ȃ ǂȂ ł u Ă ̂ŁA u ҂ ォ 60 ȏ ƕ L N w u Ă ܂ B. æ ÿ q p k n v q o È ¢. ú ô ½ æ y ì) i ± æ ú 6 Ó* 9 i 6 ¢ d ô ½ P 3 ÷.

Jsca x V b N C X g N ^ O u K Ð ł̃V J b N A V b g I g b v J b N A N G V i J b N g p A J b N ̌ Z ̃J b N c A s ƎҌ ̎ i A x V b N C X g N ^ ̎ O u K ł B. Í K J 1P !!!!!PG ú ã Ð f Æ J. €(èeight="32">€» Inôhis ChicagoÆedÌetter €@æil (s=š"€àspanŒ§›ð sup>11 €@µ le”à=¶r€àsª¨¸Dsiz¸@¹Ñsup>12 Asï“xlastˆØ“ n“@vŠ Ìdis‘*‘o‹ iŒÈ“aòepay ¢cashæŠ@Œ y”â–=ŠËåmbodi“ðŽài•0’ r‘h Xtrad”°•ë— obta ’ ˜)mŒ¸— íarke‘´—±„Ï Ô ùa—Hm•Œ ™Klikelihood™q¿™Ø’ ÅFSFš.

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