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µ 8ó2éÁÎVBê~!¼× íô ÓLÉ ©€7Ì Q9 é©Èþè9 #Óltë &Ê Š2Å› ’ìz³1嘞¤’Iäš²U ì µ8W‰Z¼ª3Ø£B4טQE ¸QE ¬ G¿z( ëý1fºKû9šËTŒmŽî óc9Ú㣦 „ñžF è7âY. µ Ü B!. Suppose that E(X)=µ, Var(X)=s2 Then (i) E(Yn)=µn (ii)If µ 6= 1, then Var(Yn)= s2µn¡1(1¡µn) (1¡µ) If µ =1 then Var(Yn)=ns 2 Proof Was given in lectures (and a different proof can be found in Notes 4) Some additional properties of conditional expectations 1 If X and Y are independent rv’s then E(XjY)=E(X) Proof.

W µ!c(ò2 $ x ± ¼ p §4ß ì!c(ò b ¾ p c>* ,¡4 x 7 ö 5 '¼ b " Ê Ü#Õ æ @ w M 5 0 ^ #Õ Â(Ô µ 6 >, Q b S u>* " Ê Ü#Õ æ b4 B í. Q U K q ß Q h { f \ p z Z p x µ » ´ Ü t S Z µ Ù À w & g A ® µ » ´ Ü t S M o z ¼ ù ± Ï µ w l q ï » å « ³ ã ï b \ q t z. 0 1 2 3 4 5 6 7 8 9) 3 /;.

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Definition 2 Let X and Y be random variables with their expectations µ X = E(X) and µ Y = E(Y), and k be a positive integer 1 The kth moment of X is defined as E(Xk) If k = 1, it equals the expectation 2 The kth central moment of X is defined as E(X − µ X)k If k = 2, then it is called the variance of X and is denoted by var(X). Ü µ ¡ K #æ3¸ v / W Z C T I 8 Ó î Ý » Þ « _#æ3¸ Õ å ­ Ý b& 1 Ó î Ý @¬ ·3æ I r M G b Ó î Ý @ ?. ½ Ü µ Ù À p w w í z ± p V D ó Q U K q Ì t z Ú ­ Â ï ¬ w » ® ¿ Ä 0 Å U b Í p ;.

} ± _)° ` M \ 0 Iy /² g!· HÍ HÕ ê ö ­ È å HÖ § § º § v P ¥ ?. } *Ë ( @)F E Z A S HÓ #Õ "@ Ø ( 0{ HÔ HÍ ¶ º ?. í î Ú Ù Æ Ç ç Ñ Ù à ¸ µ ¶ ¼ µ ¾ ¹ º Ô º ¹ ¿ ¹ Á ¼ ï ¸ ¶ ¼ ½ ¾ Ý ¹ ¶ ¸ ¾ ð ½ ¾ ¶ · í î Ú Ù Æ Ç ç ¼ µ ¿ Ô À ë ë Æ Ç ñ í î Ú Ù Æ ò ì Ñ Ù Ø Ã ¶ ¼ µ ¾ ¹ º Ô º ¹ ¿ Ñ ¸ Å ¼ ¶ ¹ Û ¸ Æ Ç.

Î ° ¸ 3 º Î ñ !. 3Û D KY Sä µ cø l!. á ¶ Å Ä · Ñ Ñ · Ý ¹ Ò Ø Ù Í Ï µ ¹ · Ò ¹ Ó Ó ´ Ô ¹ µ Ç ¼ ¹ Í ¸ Ç 5 h = > 4 0 9 2, 1?.

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( #`*>>& U µ d >' § £ R >& U µ d >' Akiko MIYAMOTO*1 Kaori SAKO*1 Rumi FURUKAWA*1 Ran MURAYAMA*1 *1 Sanki Engineering Co, LTD Micro plastics pollution is becoming a global issue and many data of that have been measured in foreign countries In Japan some data on the micro plastics pollution of rivers or sea water have been reported. CovX,Y=E(X −µ X)(Y −µ Y)=EXY−µ Xµ Y IfX tendstobelargewhenY islarge,thecovariancewillbepositive 2 If two random variables are independent, their covariance is zero However, the opposite is not (quite) true two random variables can have zero covariance without being independent 3 Thecorrelation coefficient ofX andY is ρ. 0Á "g v \ %Ê µ b 4 · _ P M 0b 7 (Ù M S u _ è W b Æ b 1* / W S >' %Ê µ _ 6õ M 1* >' 0Á "g v _ > 8 Z M g  >' 0Á "g v \ %Ê µ b 4 · _ 6õ M 0b D >, 8o % ?} ^ á å ­ µ º b ¥ å ª Ù å º ( Ò 1* & E >, ¬ #ä 0° 3Ù g ' b 0b >, m )F A \) Ý b # >' %Ê µ _ 6õ M 1* >8 £ 1* i r.

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