MIT - Massachusetts Institute of Technology

1 # #$ # ( & ( ! # ! * ! + # , # ' 2=(x )2 n 2=x2 n 2 s2& #' s2=(x x )2 n 1 s2=x2 (x )2nn 1 * ! + s ) $ " ! % sn ! ! % s21n1+1n2 & s12n1+s22n2 !"! % " " ' !" #" # " ! " ! $ # # ' " ! ! 2 43+ !" " ! " " " # " ! # " ! $ # 2 43 " " $ ' " ! ! 2 3+ !" " ! " " " ! " # 1 ! " " 1 ! !! " 4 "-!" " !" + !" ( ! " "% 4 t=x 0sn ! 2 3 > " " "- !" #" " $ # ! " -$ # 2 "' " " " ! % ! " ' $ 4 ! " # " 3 #! " # " "-!" " !" ! , + A 4 " "-!" " !" ! ' ! " $ * " !))

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

3 " $ " ' # ! + x = s2=(x x )2 n 1=( )2+( )2+ 10 7s=s2= 10 4 "-!" " !" !+ t=x 0sn= 10 47= " ! > ;-5 > : " " "- !" #" " $ # ! " ) 6,5; % " : ! ! "% 4,49 >4,469, ! ! " " " -$ # ! !! " 4,49) ! ' # " 4 # " " " ! # $ ' # " % " & ! " ", !"! % " " ! "% # " ! ! " ' " " $ # # ! ! " ' " $ # " " # * + " $ # ! " # " " " $ # " " # " " " "% $ # ! $ # " " ! " $ # ! 2" !3 #! 4 > 4 " -! "-" !" "% # " ! " 4+ !" " ! " " " ! " "% # " ! # 2 5> 63 + !" " ! " " " ! " "% # " ! # # !

4 " " " " 2 5? 6) 5A 6) 5@ 63 !+ ! ! ! ) " " # ) t=x 1 x 2s21n1+1n2 " " " # ) t=x 1 x 2s12n1+s22n2 ; . 1+1/:. 2+1/ ! ! !+ ! "! ! # " ! +# " " ! " ! !+ ! ! ! * ! ! ! !+ ! ! ! . " ! ! ! / ! " ! +# " 1< 2* ! 18 ! * 12 $ ! ! ! # # 700 & . ! # ! ;21 & /* ! 6 " ! ! ! # # 668 & . ! # ! ;30 & /* ! ! ! ! ! . " ;0*05/, 1;700' 1;21' 1;12( 2;668' 2;30' 2;6 0) 1; 2 ) 1= 2 . " $ & ! ! ! / ! " ! ! ! # ! !$ " ! " " ! ! ! " ! ! # !& * !+ ! ! ! ) t=x 1 x 2s12n1+s22n2 =700 66821212+3026=.

5 1+1/:. 2+1/ ; .12+1/:.6+1/;16 ! !+ ! "! ! ' ! +# " !$ 0*01 0*02' $ ! 0* ! ! ! ! ! * $ ! ! # !! ! ! # ! "! 0) ! # !! ! ! # ! "! ) ! # !! ! ! ! # ! "! + " ! ! ! ) 2=(O E)2E . ! # # " ! % ! # " / ; " ! ! ! "! - 1 ! ! +# " ! ! 2 ! # " " ! " ! 2 # " * ! +# " ! ' ! # ! ! ! ! % ! ! "! * = ' ! ! & ! ! % ! ! "! " " $ " ! ! ! !! ! ! " !&* " ! " ! " * " " ' " $ ! " ' " * # " ! 1 ! " * 2 $ $* ! ! " !& ! ! & . " ;0*05/, " " " " 143 60 55 18 ! ! ! !& ! 9)3)3)1' ) 0) !

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

7 # # 3 ! # % $ ") # ,% $ " ! # ! # 1+36) " & # ! # 1+ $ " " ( $ ! "#! $# + " & # ! #& # ! % ! " ! ! # ! # . / 1* # #& % ! " ! # * # #& % ! " ! # # " # ( ""$ # " $# ' # "#! $# " !% % $ " .02) 03) 04) 05/ ! $"$ ( ! " # " # + ! & " # !( % ! 2 $ " # !( % ! 3+ ! 2 # " # !( # !( # !( 02 03 02;03 ! 3 # !( 04 05 04;05 # " 02;04 03;05 02;03;04;05 ! !# # !( % ! 2 " # $ ! % $ " # !( % ( # # # $ ! % $ " #1+#3#1+#2+#3+#4 + ""$ ) # ' # $ ! % $ " # # & # # !( % ! 3 " # ! !# # !( $ # ( # $ ! % $ " # !( #1+#3#1+#2+#3+#4 #1+#2()+ $") # ' # % $ "* E=(#1+#3)(#1+#2)#1+#2+#3+#4=(rowtotal)(c olumntotal)grandtotal !)))))))))

8 " ! < . ,2/. ,2/ & ! " # $ ! ! &" " # $ ! $ " ," $ ! "# # "# " "# 2=(O E)2E # ,% $ " ! # # 3 ! # % $ "+ % # # &) " # ! ! # " #& # "" % " #" . ""$ <1+16/- # "" % & $ , & $ , % ! " 224 224 221 26: 5:6 ! ! " !" 22: 246 283 2:1 727 2 # : ! ## " ( 88 :2 97 76 42: = 21 ! ## " ( 292 263 235 84 641 5:1 5:2 5:3 598 2:71 ,& ! & ! % " # ' % # & E=(rowtotal)(columntotal)grandtotal=(495 )(490)1960= ! " ( " ( ! '31 '.2 '55 -10 -10'/- -10'2/ -1/',2 - 5 $ 35'31 35'5- 4,',4 35'.2 7 -, $ -/.'1 -/.'33 -//',0 -/-'25 2=(O E)2E =(113 ) +(113 124)2124+(110 ) + 6* (-+* (-+6*0(-+*0(-+65.)))))))

9 ! % (! ,',,-% " , " ! ' $ $ $ $ $ ! P(A and B)=P(A) P(B) " ! $ ! ! P(A or B)=P(A)+P(B) P(A and B) $ ! ! ! P(A|B)=P(A and B)P(B) P(A|B)=P(B|A) P(A)P(B) & $ $ $ " " ' " ' " ( $ " $ ( $ " ! ' $ ' $ ! $ ) $ " " (A and B)=P(2 children = bb and 1st child bb) $ " . % P(B)=P(1st child = bb) = $P(2 children = bb | 1st child bb)=P(A|B)=P(A and B)P(B)= " $ 1-2 # /-2 !' $ #& " $ " #( ) % + 3 #* 3 , ) + 3 # 3 , & $ % P(A|B)=P(B|A) P(A)P(B) P(S=Z|W=Zz)= +0-2 , P(W=Zz)= + , P(S=Z)=( )+( 1)= + # , P(W=Zz|S=Z)= " $ " " P(X=m)=n!)

10 Pm (1 p)(n m)m! (n m)! 3 " " " ( P(1 boy of 5 children)=5! ! (4)!= $ "& & P(X=m)=e np (n p)mm! % !)