probabilites - Département Informatique
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probabilites - Département Informatique
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Fm,n %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%65 &9%&% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%65 &9%/% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%66 &9%0% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%66 " : "? " B ": " / ! : A " B = ( b1,b2 + & ,b j, ,bβ ) % A " % " A = ( a1,a2, ,ai, ,aα ) B :" α " ai "" "A ( a ,b ) % i j . " P = αβ β " B / " B ": C " " /> ": " " & B ": bj % " " B% " < B λ " ( ) ( ) S = s1,s2 , ,siλ , ,sσ % A σ " S + ( ) B = b1,b2 , ,bi2 , ,bβ C α " β " ai1 C B λ " B ": " " AC " /> B ": " " S% " " Bλ " ×σ "& bi2 C %%%C " " C " ( a ,b , i1 " " BC %%%C " λ > B ": . Mλ " FC siλ % "" Mλ = αβ × AC BC %%%C S :" " A = a1,a2 , ,ai1 , ,aα C " / i2 ) ,siλ % " "E + 0 " C E = ( a,b, ,s ) % n " p " "" p : " " C C = ! :B p " n " " n " " ' "? ? % . Anp " Anp = np : p " nC " C p " "E + n " 1 " C E = ( a,b, ,s ) % p " "" p : n " " n " % . A pn " A pn = n! (n − p )! 1≤ p ≤ n : " C " C " "E + " C E = ( a,b, ,s ) % n " n " "" ": 4 " n " ' " C "% . Pn " Pn = n! C C " " " E n " 5 v " " C E = a,a, ,a , b,b, ,b , , s,s, ,s groupe1 groupe 2 avec α lettres a avec β lettres b + " n " '"" a α C C %%%C " " % . Pv " Pv = n! α! β! × × σ! + σ = n% groupe v avec σ lettres s n " "" ": α +β+ :" C " s σ " "E + n " 6 " C E = ( a,b, ,s ) % p " "" p : n " " n " . Cpn Cpn = n p n! p! ( n − p ) ! 1≤ p ≤ n " C " % C " "E + 7 " C E = ( a,b, ,s ) % n " p " "" p : " "C " ? % . Cnp " Cnp = Cpn+p−1 = n,p ≥ 1 C " (n + p − 1)! p! ( n − 1)! n " " n " " C ' "? p C = ! :B p " "E : "E n " ": 8 " C E = ( a,b, ,s ) % " {∅,a,b, ,s,{a,b}, ,{r,s} ,{a,b,c}, ,E} % . Card (P (E ) ) " Card (P (E ) ) = 2n P (E ) P (E ) ": " EC " " " " " ; ; " "C ": : = &9 > "! " " ! " % % !"#$%&'(&) *+#*,-%$&) &,#"$&./&) > G ? ! "% " %( < ; "C = % " ! <" ! ! "<; ; B <; < " ! " % " B "< " " " % ! % ( > ; ; ; C > C "C C " !< = " B ; % % #).+,*,) 01.'&&!"#$%&'(&&,01.'&#"$&./& "< " ! " " "<; % Ω% ; " " " "C " = ' Ω = {1,2,3,4,5,6} % < " Ω G ; "" C < " ! "" C ; "<; "" " ' H " % " B ; C " C " " "< && " Ω% !,&')%-' 0&+*'-,%-' 01&!"#$%&'(& ; < " "Ω % " i " /> " " =C" " % ; " ( i,j ) "C / = ! "& = " j' Ω = {(1,1) , (1,2 ) , , (1,6 ) , ( 2,1) , , ( 6,6 )} % 23&(,%40.(*+(.+0&) "$-2*2%+%,#) ( > ; " " < % < " " " C < B " " " % " " < ! " ! "< " " &/ !,&')%-' 0&+*'-,%-' 01#/#'&5&', < " " ; B ; " B "C "< " A = {2,4,6} $ " " ΩC "< Ω% A < C $ I "! " ! I" = " " Ω = {1,2,3,4,5,6} % "< -$$&)"-'0*'(& &',$& +& +*'6*6& 0&) #/#'&5&',) &, +& +*'6*6& &')&52+%),& 3 ) )I ) AI A A , A A ∩B I( ,I I( ,I A ∪B . IΩI Ω " " I∅ I Ω $ " I(I Ω " < ! " I(I I( ' A∪A =Ω " " ,I I,I ( " "! I,I , ' A ∩B = ∅ = A ⊂B " $%2. Ω "< P ( Ω ) "< Ω @ " Ω " CA " A% " $ '+ < Ω " B $ "! " ! Ω % " P (Ω) % P (Ω) " @ P (Ω) % Ω '+ " B " A &0 " "" < " " ∅ ∈ A et Ω ∈ A A∈A A∈A A1,A 2 , ∞ n=1 An ∈ A < ∞ n=1 An ∈ A % AC " " ! " &1 1 Ω "< " J (; P(A) C AC < A ! < "< ; ΩC " A " " " " " " ; < " " & 0 ≤ P(A) ≤ 1 (; / P (Ω) = 1 (; 0 A1,A 2 , ,A n P n∈N (; An = n∈N < n=1 An = C " " A1,A 2, P = P ( An ) 1 ∞ /B/ < ∞ n=1 P ( An ) /B/ = C " B " " Ω "< Ω CA " A " ; &C /C 0 P " % "< &4 " " " 1 K0 C " " " " ( Ω,A ) % " " " ( Ω,A,P ) Ω " &5 -%) 0&"$-2*2%+%,#) 0%)($7,&) Ω " P " > " " < " " % )"*(&4%'% n " Ω C Ω = {w 1,w 2, ,w n} % wi < " / "<; 0 ! P(A) = P ( wi ) % " w i∈A P ( wi ) " (; " P " A = > " pi " C " " " " & 0 ≤ P ( wi ) ≤ 1 (; n i=1 / P ( wi ) = 1 )"*(&%'4%'%0#'-52$*2+& " Ω < " P(A) = " w i∈A P ( wi ) % G A w1,w 2, / " < = C " " " "<; 1 " " P " > P ( wi ) " (; &6 " " pi " " & 0 ≤ P ( wi ) ≤ 1 (; ∞ i=1 / P ( wi ) = 1 )"*(&8.&+(-'8.& + " " "Ω! " ! % > $ " "% -%) 0&"$-2*2%+%,#) (-',%'.&) " " " ""% P Ω " " " " &7 #4%'%,%-' " " " " " < ; A " A < " Ω B A ∩B% P ( A ∩ B) P (B ) P (B ) ≠ 0 "" P ( A|B ) " (; & 0 ≤ P ( A|B ) ≤ 1 (; / P ( Ω|B ) = 1 (; P 0 n i=1 A i|B = n i=1 P ( A i|B ) " " ! "< "" P ( A|B ) " P ( A|B ) = B ! " "< $= "" < P ( A|B ) C " C "< ( Ω,A,P ) ; % B " &8 #0.(,%-' 0&+*4-$5.+&0&) "$-2*2%+%,#) (-5"-)#&) / ( , P ( A ∩ B ) = P ( A ) × P (B|A ) = P (B ) × P ( A|B ) 0 (C ,C P ( A ∩ B ∩ C ) = P ( A ) × P (B|A ) × P ( C|A ∩ B ) A1,A 2, ,A n P ( A1 ∩ A 2 ∩ ∩ A n ) = P ( A1 ) × P ( A 2|A1 ) × P ( A 3 |A1 ∩ A 2 ) × P ( A n|A1 ∩ × ∩ A n−1 ) #0.(,%-' 0&+*4-$5.+&0& *9&) / P ( A|B ) = ( , P ( A ) × P (B|A ) ( ) ( P ( A ) × P (B|A ) + P A × P B|A ) P(A) + P(A) = 1 ; C B% P ( A ) "< " A " B A1,A 2, ,A n P ( A i|B ) = P ( A i ) × P (B|A i ) n i=1 n i=1 P ( A i ) × P (B|A i ) P ( Ai ) = 1 A > "< "" " " B ,? % A P ( A|B ) C " " /9 1 #4%'%,%-' 0&+1%'0#"&'0*'(&0& #/#'&5&',) " A ( Ω,A,P ) B $= <" B ; " B "< "< % / " P ( A ∩ B ) = P ( A ) × P (B ) P(A) > 0 P (B ) > 0 C P ( A|B ) = P ( A ) < ! P (B|A ) = P (B ) / < " " ! " " "< " " "< % #4%'%,%-' 0&+1%'0#"&'0*'(&0&"+.)%&.$) #/#'&5&',) " ( Ω,A,P ) $= B A1,A 2, ,A n nC j = 2,3, ,n C ( P A i1 ∩ A i2 ∩ " C P ( A1 ∩ A 2 ∩ ; $ " ) ( ) ( ) ∩ A ij = P A i1 × P A i2 × ( ) × P A ij j = nC ∩ A n ) = P ( A1 ) × P ( A 2 ) × × P ( An ) " " A i1 ,A i2 , ,A ij % n j " " < " " ! B " "" < % % /& " R% < $B$ "X ( w ) ! C " ! " ! C X C " w "<; " " " " " % " " : // 1 #4%'%,%-' "Ω + "" "< " X:Ω → R % ! < "" I " I R X−1 (I) C XC " X X−1 (I) = {w; ∃x ∈ I / X ( w ) = x} I = R, X−1 (I) = Ω I = ∅, X−1 (I) = ∅ -,*,%-') . C X(w) = X X −1 $& D ( ]x,y ]) = ( x < X ≤ y ) X−1 ( ]−∞,x ]) = ( X ≤ x ) Ω $& D D ; L X−1 ( ]x,∞[ ) = ( X > x ) X−1 ( x ) = ( X = x ) 5*6&$#(%"$-8.&01.'&$#.'%-' &,01.'&%',&$)&(,%-' 01%',&$/*++&) I1,I2, X−1 ∞ i= 0 < Ii = " ∞ i=0 X−1 (Ii ) ! X−1 ∞ i= 0 < "" % < ! Ii = < ! ∞ i=0 "" % "" RC " X−1 (Ii ) "" < < " ! < "" "< " /0 #4%'%,%-' " ( Ω,A ) " X " " ! "< "" "" X: ( Ω,A ) → R % "" I % % RC "< B" ∀I ∈ R, X−1 (I) ∈ A % *$%*2+&*+#*,-%$&0%)($7,& X % % > " X (Ω) "< "% *$%*2+&*+#*,-%$&(-',%'.& X % % " X (Ω) "< "" R B % !&5"+&) *$%*2+&*+#*,-%$&(-'),*',&;-.(&$,*%'&< *$%*2+&*+#*,-%$&%'0%(*,$%(&;-.0& &$'-.++%< $-2*2%+%,#%5*6& " p ( ∀I ∈ R, p (I) = P X−1 (I) ) ( Ω,A,P ) % %X ( Ω,A ) C " " " " %p " /1 P " P K0 /% p " (; " ; " & 0 ≤ p ( I) ≤ 1 (; / p (R ) = 1 (; 0 ∞ p i=0 Ii = 1 ∞ i= 0 p (Ii ) -'(+.)%-' p ( I) C " Ω D P ( I) C " > $& " ! <"" . $& " D " " & " % % X " $& B ; " M D B " "< 9 ( Ω,A,P ) % " P ( I) > " I = ]−∞,x ] C < $B$ " P( X ≤ x) B "< " " B "< X−1 ( ]−∞,x ]) = ( X ≤ x ) % X≤x % "" " = /4 1 #4%'%,%-' + F:R → R [0,1] " % % XC " "" "" ! F(x) = P( X ≤ x) % % $-"$%#,#) & 0 ≤ F(x) ≤ 1 / F(x) 0 lim F ( x ) = 1 x →∞ 1 lim F ( x ) = 0 x →−∞ 4 F(x) B D' F ( x ) = F ( x + ) > % " /5 *+(.+0&"$-2*2%+%,#) */&(+*4-'(,%-' 0&$#"*$,%,%-' @ F(x) " " % % X < " ! X C " ""% & P ( X > x) = 1− F ( x) / P ( x1 < X ≤ x 2 ) = F ( x 2 ) − F ( x1 ) 0 B 2' P ( X < x ) = F ( x − ) P( X < x) 1 P ( X = x ) = F ( x+ ) − F ( x− ) ! " F xC " " " F ( x + ) = F ( x − ) =F ( x ) B F < 1' % % < < % % > P( X = x) = 0 ""% F ( x+ ) > F ( x− ) xC " B % % % % > !&5"+&01.'&4-'(,%-' 0&$#"*$,%,%-' P( X = x) > 0 ""% " /6 =-'(,%-' 0&$#"*$,%,%-' &,+-%0&"$-2*2%+%,#) 01.'&/*$%*2+&*+#*,-%$& 0%)($7,& $-2*2%+%,#) *,,*(>#&) ? .' "-%', &, ? .' %',&$/*++& 01.'& /*$%*2+&*+#*,-%$&0%)($7,& *""&+0&+*0#4%'%,%-' 01.'&/*$%*2+&*+#*,-%$&0%)($7,& X % % > "< " x " + " " X (Ω) X "% x " "C < $B$ % $-2*2%+%,# *,,*(>#& ? .' "-%',01.'& /*$%*2+& *+#*,-%$& 0%)($7,& < >" 1 K1%0 0 P ( X = x ) = F ( x+ ) − F ( x− ) > 0 " B < % % > ""% $-2*2%+%,# *,,*(>#& ? .' %',&$/*++& 01.'& /*$%*2+& *+#*,-%$&0%)($7,& < >" 0 K1%0 0 P ( xi < X < xi+1 ) = P ( X < xi+1 ) − P ( X ≤ x i ) = F ( xi−+1 ) − F ( x i ) = F ( xi ) − F ( xi ) = 0 " B "" < % % > ""% =-'(,%-' 0&$#"*$,%,%-' F(x) = P( X ≤ x) 4 K1%/ 0 " " < > " K1%4%&%/ &C /C 0C 1 1%4%&%0 0% " /7 -%0&"$-2*2%+%,#) P( X = x) (; " " " " ; C " & 0 ≤ P( X = x) ≤ 1 (; x / P( X = x) = 1 &+*,%-' &',$&+*4-'(,%-' 0&$#"*$,%,%-' &,+*+-%0&"$-2*2%+%,#) F(x) = xi ≤ x P ( X = xi ) %),$%2.,%-' 01.'&/*$%*2+&*+#*,-%$&0%)($7,& < " " % % > G P( X = x) N ! " F(x) = P( X ≤ x) % =-'(,%-' 0& $#"*$,%,%-' &, 0&')%,# 0& "$-2*2%+%,#) 01.'& /*$%*2+& *+#*,-%$&(-',%'.& $-2*2%+%,#) *,,*(>#&) ? .' "-%', &, ? .' %',&$/*++& 01.'& /*$%*2+&*+#*,-%$&(-',%'.& *""&+0&+*0#4%'%,%-' 01.'&/*$%*2+&*+#*,-%$&(-',%'.& X % % "< " X (Ω) "" R B " /8 % $-2*2%+%,# *,,*(>#& ? .' "-%',01.'& /*$%*2+& *+#*,-%$& (-',%'.& < >" 1 ( ) K1%0 0 ( ) P ( X = x) = F x+ − F x− = 0 " B < % % ""% $-2*2%+%,# *,,*(>#& ? .' %',&$/*++& 01.'& /*$%*2+& *+#*,-%$&(-',%'.& < > " K 1%5%&%/C " ]a,b] " [a,b[ C B ]a,b[ "% [a,b] C "" < >" / $ K1%0 0 P ( a ≤ X ≤ b ) = P ( a < X ≤ b ) = P ( a ≤ X < b ) = P ( a < X < b ) = F (b ) − F ( a ) " B " "" < % % " "B" ; ; "< ""% =-'(,%-' 0&$#"*$,%,%-' F(x) = P( X ≤ x) 4 K1%/ 0 " < > " K1%4%&%/ &C /C 0C 1 1%4%&%0 0% &')%,#0&"$-2*2%+%,#) " " % C " " " B C "" " "" [a,b] "" "% " " B "" " B " < " B F ( b ) − F ( a ) = d F ( x ) C < $B$ " F ( x ) % + C d F ( x ) = F′ ( x ) dx % ( "" % % 09 G " F ( x ) % F′ ( x ) F′ ( x ) " f (x) C " " " " % & F′ ( x ) = f ( x ) ⇔ F ( x ) = x f ( t ) dt −∞ / f (x) ≥ 0 0 +∞ f ( x ) dx = 1 −∞ 1 b P ( a ≤ X ≤ b ) = F ( b ) − F ( a ) = f ( x ) dx a &+*,%-' &',$& +* 4-'(,%-' 0& $#"*$,%,%-' &, +* 0&')%,# 0& "$-2*2%+%,#) F(x) = x f ( t ) dt −∞ %),$%2.,%-' 01.'&/* (-',%'.& < " % % f (x) G N ! " F(x) = P( X ≤ x) " 0& 1 #4%'%,%-' < ? " ! % % " " " " / " ! "% < E ( X) ! % % X < > % % % % % )"#$*'(&01.'&/* 0%)($7,& E ( X) = x xP ( X = x ) E ( X) ; " < E ( X) ! % F(x) A; & ; P ( X = xi ) ; P ( X = x2 ) ; P ( X = x1 ) ; x1 x2 9 xi ; )"#$*'(&01.'&/* (-',%'.& E ( X) = +∞ −∞ xf ( x ) dx " E ( X) ; "< " < E ( X) ! 0/ % F(x) Fonction de Distribution de Probabilité p=inormal(x;0;1) 1,0 0,8 d[F(x)]=f(x)dx 0,6 0,4 0,2 0,0 -3 -2 -1 0 x 1 $-"$%#,#) & % %X / a bC " E ( aX + b ) = aE ( X ) + b / a C " E (a) = a 0 "; E ( X) ≥ a a "" ! P ( X ≥ a ) = 1C " 2 3 " 1 b "" ! P ( X ≤ b ) = 1C " "; E ( X) ≤ b 4 "; / a b "" ! P ( a ≤ X ≤ b ) = 1C " a ≤ E ( X) ≤ b 5 P ( X ≥ a) = 1 E ( X) = a C " P ( X = a ) = 1 et P ( X > a ) = 0 P ( X = E ( X ) ) = 1 et P ( X > E ( X ) ) = 0 00 " @ 01 1 #4%'%,%-' " "< "< $ ? ( V ( X) = E X − E ( X) $? " " / ! % %X < < ! 2 ) ( ) = E X2 − E ( X ) % V ( X) "B σ ( X) "B 2 % %X < % % σ ( X) = V ( X) V ( X) " ? B" V ( X) % < ? G " E ( X) % " *$%*'(&01.'&/* 0%)($7,& ( V ( X) = E X − E ( X) ( ) 2 ) ( x − E ( X)) P ( X = x ) 2 = x V ( X ) = E X2 − E ( X ) = x 2P ( X = x ) − E ( X ) 2 x 2 *$%*'(&01.'&/* (-',%'.& ( V ( X) = E X − E ( X) ( ) − E ( X) V ( X) = E X 2 % 2 2 )= = +∞ −∞ +∞ −∞ ( x − E ( X ) ) f ( x ) dx 2 x 2f ( x ) dx − E ( X ) 2 < " 04 &5*$8.&).$+&(*+(.+0&+*/*$%*'(& "C " " ( ) " " " " " V ( X ) = E X2 − E ( X ) C ! 2 % % > " " % $-"$%#,#) " B "< & V ( X) ≥ 0 / % %X ; V ( aX + b ) = a2 V ( X ) 0 a C " V (a) = 0 1 % % XC " V ( −X) = V ( X) 4 P ( X = a) = 1 " V ( X) = 0 a bC " "< % " 05 #4%'%,%-' + " " " Bn " " " " ! "" "< " n % %% " ( Ω,A ) X: ( Ω,A ) → Rn % X % % Bn "" % % B n : "" I Rn C "< B" ∀I ∈ Rn, X−1 (I) ∈ A X−1 (I) = {w; ∃ x1 ∈ I1, , ∃ xi ∈ I,i , ∃ xn ∈ In / X1 ( w ) = x1, , Xi ( w ) = xi, , Xn ( w ) = xn } . X(w) " ( X ( w ), 1 ,Xi ( w ) , ,Xn ( w ) ) I " (I1, ,Ii, ,In ) % =-'(,%-' 0&$#"*$,%,%-' 0 .'&/*$%*2+&*+#*,-%$&?"+.)%&.$) 0%5&')%-') F ( x1, ,x i, ,x n ) = P ({ X1 ≤ x1} ∩ ∩ { Xi ≤ x i } ∩ = P ( X1 ≤ x1, ,Xi ≤ xi, ,Xn ≤ x n ) ∩ { Xn ≤ x n } ) & 0 ≤ F ( x1, ,x i, ,x n ) ≤ 1 / F ( x1, ,xi, ,xn ) B " xi % " 06 0 lim F ( x1, ,x i, ,x n ) = 1 xi →∞ i=1, ,n 1 lim F ( x1, ,xi, ,xn ) = 0 xi →−∞ i=1, ,n 4 B ' F ( x1, ,x i, ,x n ) = F ( x1+ , ,xi+ , ,x n+ ) F ( x1, ,xi, ,xn ) -% 0& "$-2*2%+%,#) 0 .'& /*$%*2+& *+#*,-%$& 0%)($7,& ? "+.)%&.$) 0%5&')%-') P ( X1 = x1, ,Xi = xi, ,Xn = x n ) " " " " (; & 0 ≤ P ( X1 = x1, ,Xi = x i, ,Xn = x n ) ≤ 1 (; x1 / xi xn P ( X1 = x1, ,Xi = xi, ,Xn = x n ) = 1 F ( x1, ,x i, ,x n ) = x1j ≤ x1 xi j ≤ xi xn j ≤ xn ( P X1 = x1j , ,Xi = x ij , ,Xn = xn j ) " 07 &')%,# 0& "$-2*2%+%,#) 0 .'& /*$%*2+& *+#*,-%$& (-',%'.& ? "+.)%&.$) 0%5&')%-') F ( x1, ,x i, ,x n ) = x1 xi xn −∞ −∞ −∞ f ( t1, ,t i, ,t n ) dt1 dt i dt n & ∂nF ( x1, ,xi, ,xn ) f ( x1, ,x i, ,x n ) = ∂x1 ∂xi ∂xn "" : + B " B " "= C : $B$ "% / f ( x1, ,x i, ,x n ) ≥ 0 0 +∞ +∞ −∞ −∞ +∞ f ( x1, ,x i, ,x n ) dx1 dxi dxn = 1 −∞ 1 P ( a1 ≤ X1 ≤ b1, ,ai ≤ Xi ≤ bi, ,an ≤ Xn ≤ bn ) = b1 bi bn a1 ai an f ( x1, ,x i, ,x n ) dx1 dxi dxn " " " 08 *$%*2+&) *+#*,-%$&) %'0#"&'0*',&) #4%'%,%-' % % X1, ,Xi, ,Xn F ( x1, ,x i, ,x n ) = P ( X1 ≤ x1, ,Xi ≤ xi, ,Xn ≤ x n ) = P ( X1 ≤ x1 ) × = F ( x1 ) × × P ( Xi ≤ x i ) × × F ( xi ) × n % % × P ( Xn ≤ x n ) n × F ( xn ) = ∏ F ( x i ) i=1 % %Bn "% *$%*2+&*+#*,-%$&) 0%)($7,&) %'0#"&'0*',&) : >" n P ( X1 = x1, ,Xi = xi, ,Xn = x n ) = ∏ P ( Xi = xi ) i=1 n n i=1 i=1 F ( x1, ,x i, ,x n ) = ∏ P ( Xi ≤ xi ) = ∏ F ( x i ) *$%*2+&*+#*,-%$&) (-',%'.&) %'0#"&'0*',&) : >" n F ( x1, ,x i, ,x n ) = ∏ xi i=1 −∞ n f ( t i ) dt i = ∏ F ( xi ) f ( x1, ,xi, ,xn ) = ∏ f ( x i ) i=1 n i=1 " " 19 )"#$*'(&0&"+.)%&.$) /*$%*2+&) *+#*,-%$&) & n % %C " X1, ,Xi, ,Xn n E i=1 Xi = n i=1 E ( Xi ) / n % %C a1, ,ai, ,an X1, ,Xi, ,Xn n E i=1 a i Xi + b = n i=1 bC C " aE i ( Xi ) + b 0 n % % X1, ,Xi, ,Xn E n ∏X i=1 C " n i = ∏ E ( Xi ) i=1 1' " / % % Cov ( X1,X2 ) = E ( X1X2 ) − E ( X1 ) E ( X2 ) 4 X1 X2 C " E ( X1X2 ) = E ( X1 ) E ( X2 ) Cov ( X1,X2 ) = 0 *$%*'(&0&"+.)%&.$) /*$%*2+&) *+#*,-%$&) " B ": ": % " 1& & X1, ,Xi, ,Xn n V i=1 n Xi = i=1 n % % C " n % % C a1, ,ai, ,an V ( Xi ) / X1, ,Xi, ,Xn n V i=1 ai X i + b = n i=1 b ai2 V ( Xi ) 0 X1 ( X2 "" ! E ( X1 ) = E ( X2 ) C " / % % E [ X1 − X2 ] 2 ) = V (X ) + V (X ) 1 2 C " " 1/ = #4%'%,%-' X "tC % % ΨX ( t ) " "" ! ( ) ΨX ( t ) = E etX E ( X) ; K4%& ΨX ( t ) ; % =-'(,%-' 6#'#$*,$%(& 0&) 5-5&',) "-.$ .'& /*$%*2+& *+#*,-%$& 0%)($7,& ( ) ΨX ( t ) = E etX = x etxP ( X = x ) =-'(,%-' 6#'#$*,$%(& 0&) 5-5&',) "-.$ .'& /*$%*2+& *+#*,-%$& (-',%'.& ( )= ΨX ( t ) = E e tX +∞ etx f ( x ) dx −∞ $-"$%#,#) & ( ) ΨX( ) ( 0 ) = E Xn n ! ΨX′ ( 0 ) = E ( X ) > " 9 " B " "< 10 % ( ) ΨX′′ ( 0 ) = E X2 " ΨX′ ( 0 ) V ( X) : " B E ( X2 ) % 9 ΨX′′ ( 0 ) " " ": E ( X) " % % X% / X ΨX ( t ) % % C " ΨaX+b ( t ) = ebt ΨX ( at ) 0 X1, ,Xi, ,Xn n % % ΨX1 ( t ) , ,ΨXi ( t ) , ,ΨXn ( t ) Ψn i =1 n Xi ( t ) = ∏ ΨX ( t ) i=1 i C " a bC / " 11 B(p) #4%'%,%-' ; p ! q = 1− p % / X" % %! " A " "B& A A " CB9 " P ( X = 1) = p P ( X = 0) = q = 1− p " P ( X = x ) = p x q1−x x = 0,1 )"#$*'(&A/*$%*'(&&,4-'(,%-' 6#'#$*,$%(&0&) 5-5&',) E ( X ) = p C V ( X ) = pq C Ψ ( t ) = p et + q " A " 14 B(n,p) #4%'%,%-' ;.'&6#'#$*+%)*,%-' 0&+*+-%0& &$'-.++%< ; ! q = 1− p % p A / " A " X" % %! A x " n " " "> % " ! A n−x " qn− x " x % > " px C % > A " ! " ! % A > " n−x C " " B p x qn− x X=x x " nC Cnx % " P ( X = x ) = Cnx p x qn− x " , x = 0,...,n " X G n > p' X = n i=1 B ( p ) = B (1, p ) = C1xp x q1− x = p x qn− x Xi x = 0,1 " , "" Xi " B(10,1/ 4) " A )"#$*'(&A/*$%*'(&&,4-'(,%-' 6#'#$*,$%(&0&) 5-5&',) ( E ( X ) = np C V ( X ) = npq C Ψ ( t ) = p et + q ) n 15 " 16 M(n,p,k) #4%'%,%-' ;.'&6#'#$*+%)*,%-' 0&+*+-% %'-5%*+&< k ; p1, ,pi, ,pk % " A1, ,A i, ,A k " Xi " % % ! xi " n Ai " " "> % " ! Ai " xi > k " k Ai " ! " B pixi C " C "B ∏p i=1 xi n= k i=1 xi C % xi i : $B$ " n! k ∏x ! i=1 k i=1 pi = 1% " P ( X = x ) = P ( X1 = x1, , Xi = xi, , Xn = x n ) = k ∏x ! i=1 k i=1 xi = n k i=1 k n! i ∏p i=1 pi = 1% )"#$*'(&&,/*$%*'(& E ( X) " " E ( Xi ) V ( X) " " V ( Xi ) xi i i J " 17 G(p) : #4%'%,%-' ; ! q = 1− p % p A " &> / " X" % %! A " x " " " : : "> x = 0C "? A " ( % A B " &> " " " " q " P ( X = 0) = p % x = 1C "? A B " &> " A B " /> " " P ( X = x ) = pqx x = 0, " " pC P ( X = 1) = pq % pC " 18 G(1/ 4) " A )"#$*'(&A/*$%*'(&&,4-'(,%-' 6#'#$*,$%(&0&) 5-5&',) E ( X) = q q p C V ( X) = 2 C Ψ ( t ) = p p 1 − qet 0 < qe t < 1 " 49 B(r,p) #4%'%,%-' ;.'&6#'#$*+%)*,%-' 0&+*+-% #-5#,$%8.&< ; ! q = 1− p % p " r > A / X" % %! " " r+x A " A x " "> % " ! " x A " ! A " " A " " r % " B pr C X=x ! " B qx C % " B pr qx " r −1 r + x − 1C Crr −+1x−1 % " P ( X = x ) = Crr −+1x −1 pr qx ", x = 0, ") G X > p' X = r r i=1 G ( p ) = B (1,p ) = Cox p qx = p qx Xi % " 2 ! Xi " B(5,1/ 4) " A )"#$*'(&A/*$%*'(&&,4-'(,%-' 6#'#$*,$%(&0&) 5-5&',) rq rq p E ( X) = C V ( X) = 2 C Ψ ( t ) = p p 1 − qet r 4& " B 4/ H(N,n,p) : #4%'%,%-' N q = 1− p " " C n ≤ N% " n " " X" % %! " " " " " x " " "> " % ) " : n " ) " : x " ": %n " " p N " " Np " " ' CNn ": " " x % ' CNp ) " " ": n−x : n− x ' CNq % " P( X = x) = x n− x CNp × CNq CNn x = 0, ,n )"#$*'(&&,/*$%*'(& E ( X ) = np C V ( X ) = N−n npq N −1 " " " Nq " " 40 U(n) = #4%'%,%-' X" % % " " 1,2, ,n " " 1 % n " P( X = x) = 1 n x = 1, ,n U(10) " A )"#$*'(&A/*$%*'(&&,4-'(,%-' 6#'#$*,$%(&0&) 5-5&',) E ( X) = n +1 n2 − 1 1 n tx C V ( X) = C Ψ (t) = e 2 12 n x =1 " P(λ ) #4%'%,%-' " e −λ λ x P( X = x) = x! x = 0, P(10) " A )"#$*'(&A/*$%*'(&&,4-'(,%-' 6#'#$*,$%(&0&) 5-5&',) E ( X ) = λ C V ( X ) = λ C Ψ ( t ) = eλe −λ t 41 " 44 U(a,b) = #4%'%,%-' X " 1 a≤ x≤b f (x) = b − a 0 autrement x<a 0 F(x) = x−a a≤x≤b b−a 1 x>b U(1,11) " " " A )"#$*'(&A/*$%*'(&&,4-'(,%-' 6#'#$*,$%(&0&) 5-5&',) (a − b) b+a E ( X) = C V ( X) = 2 12 2 45 " 46 E(λ ) C #4%'%,%-' " ; f (x) = λe −λx > λ x ≥ 0, λ > 0 0 F(x) = "" X autrement x<0 0 1 − e−λx x ≥ 0, λ > 0 E(1) " A )"#$*'(&A/*$%*'(&&,4-'(,%-' 6#'#$*,$%(&0&) 5-5&',) E ( X) = 1 1 λ C V ( X) = 2 C Ψ ( t ) = λ λ λ−t " " 47 <N(µ, σ) ; #4%'%,%-' "X ") µ > σ ? $? " 1 x −µ σ − 1 e 2 2π σ f (x) = F(x) = x 2 − ∞ < x < +∞, − ∞ < µ < +∞, σ > 0 f ( t ) dt −∞ " I= +∞ f (x) d x = −∞ A +∞ −∞ 1 e 2πσ − 1 x −µ 2 σ dx x−µ "' y = % σ " G 2 " "C N(3,0.5) " I2 % C C" e 1 − y2 2 " % " 48 A )"#$*'(&A/*$%*'(&&,4-'(,%-' 6#'#$*,$%(&0&) 5-5&',) E ( X) = µ C V ( X) = σ C Ψ ( t ) = e 2 1 µt + σ2 t 2 2 $-"$%#,#) & X % %! % % Y = aX + b a : $? " ) " ? a≠0 b µ : " ) " $? σC " " aµ + b ? aσ X ~ N ( µ, σ ) " Y = aX + b ~ N ( aµ + b,aσ ) / n % % X1, , Xn : $ ? σi C " n : $? i=1 Xi " % % Y= n i=1 " ) Xi ~ N ( µi, σi ) " Y= i=1 Xi ~ N n i=1 µi ? n Xi " ) " ? i=1 σi2 n " µi , n i=1 σi2 µi " 59 0 X1, : n % % , Xn $? σi C " Xi " % % Y= " ? i=1 Xi ~ N ( µi, σi ) " ai Xi + b a1, i=1 n " ) n Y= n i=1 " ) : ai ≠ 0 $? i=1 n i=1 aiµi + b, n i=1 µi ? b ,an n aiµi + b ai Xi + b ~ N " ai2σi2 a i 2 σi 2 1 X1, : : n % % , Xn $? $? σC " 1 " % % Xn = n n i=1 " ) Xi " " ) " σ2 n Xi ~ N ( µ, σ ) " % % X1, Xi , Xn 1 Xn = n σ2 Xi ~ N µ, n i=1 n "" : " % µ ? ? µ " 5& -% -$5*+& &',$#& #0.%,&N ( 0,1) #4%'%,%-' " ) " N ( µ, σ ) ? ") N ( 0,1) . " µ=0 F(x) = D " 1 − 21 x2 e 2π f (x) = $? . $? x f ( t ) dt −∞ N(0,1) " −∞ < x < +∞ " "" ) " σ = 1% > µ=0 ? σ =1 " 5/ A )"#$*'(&A/*$%*'(&&,4-'(,%-' 6#'#$*,$%(&0&) 5-5&',) E ( X ) = 0 C V ( X ) = 1C Ψ ( t ) = e 12 t 2 &+*,%-' &',$& /*$%*2+& -$5*+& &,/*$%*2+& -$5*+& &',$#& #0.%,& X % %! % % Y= B$ " ) X−µ σ * " ) " ) X ~ N ( µ, σ ) " Y= " " " ? ? µ=0 µ : : $? $? σC " σ = 1C : $ . X−µ ~ N ( 0,1) σ ,.0&0&+*0&')%,#0 .'&/*$%*2+& -$5*+& &',$#& #0.%,& f (x) = 1 − 21 x2 e 2π f (x) > 0 " " f (x) 50 ' f ( x ) = f ( −x ) lim f ( x ) = lim f ( x ) = 0 x →−∞ x →+∞ f (x) ; +∞ −∞ x 1 ~ ( 0,0.4 ) 2π x = 0,f ( 0 ) = 1 − 21 x2 e dx = 1 ⇔ 2π +∞ e 1 − x2 2 dx = 2π f ( x ) " −∞ f ( x ) dx = P ( −∞ ≤ X ≤ x ) = P ( X ≤ x ) −∞ = +∞ f ( x ) dx = P ( − x ≤ X ≤ +∞ ) = P ( X ≥ − x ) 1 1%5%0 −x P ( X ≤ x ) = F ( x ) = P ( X ≥ −x ) = 1 − F ( −x ) & 1%0 ,.0& 0& +* 4-'(,%-' 0& $#"*$,%,%-' 0 .'& /*$%*2+& -$5*+& &',$#& #0.%,& F(x) " . e ; 1 − x2 2 G F(x) " " F(x) ! " ! " " " ! X ~ N ( 0,1) ; P ( X ≤ 2.33 ) = 0.99 % " ") % F(x) = P( X ≤ x) % " ! " "% ! ! " " x "C F ( x ) = 0.99 "C " " " " " ) P ( X ≤ −x ) = F ( −x ) = P ( X ≥ x ) = 1 − F ( x ) % " " < $B$ X ~ N ( 0,1) x% " x : . x = 2.33 C F(x) " " " " "N " F(x) x -x' " 51 ,%+%)*,%-' 0&+*,*2+&0&F ( x ) 0& X ~ N ( 0,1) &' ),*,%),%8.&) ! :? C ; " > C ": ]−∞, x ] "" 1− α O α " % α ! ": x > " " " [ −x,x ] " ; " :? B : G ]−∞, x ] "" :? "" > "% / " " [ −x,x ] " "% ',&$/*++&0&(-'4%*'(&]−∞, x ] ( "α = 1% C x = 2.33 ( "α = 5% C x = 1.64 ',&$/*++&0&(-'4%*'(&[ −x,x ] " x → −∞ α : " x → +∞ "α = 1% C x = 2.58 ( "α = 5% C x = 1.96 " 1− α C" α " α 2 "" ]−∞, x ] " x B [ −x,x ] x "" α % "; 2 B 1− α x " B α 2 ! " B 1− α % "" G B 1− α + : " ( [ −x,x ] / " "" α ! :" ]−∞, x ] ! ": " " " % " " 54 *+-% -6@ -$5*+&LN ( µ, σ, x 0 ) #4%'%,%-' " "X $) x " > . > Y= µ " B " σ µ > "B" µ=0 σ = 1% X x0 $? " ln ( X − x 0 ) " ) " ln ( X − x 0 ) − µ σ C σ ? ") " " Y " X = x 0 + eµ+σY ~ LN ( µ, σ,x 0 ) " $) "X > µ ? C σ $? " 1 e 2πσ ( x − x 0 ) f (x) = 0 F(x) = x − 1 ln( x − x 0 ) −µ 2 σ 2 x ≥ x 0 , − ∞ < µ < +∞, σ > 0 autrement f ( t ) dt −∞ LN ( 0,1,0 ) " x0 " A )"#$*'(&&,/*$%*'(& E ( X ) = x0 + e 1 µ+ σ2 2 ( ) C V ( X ) = 1 − e−σ e 2 ( 2 µ+σ2 ) -% %'-5%*+&; 0%5&')%-')<&, .+,%'-5%*+&;' 0%5&')%-')< " % 55 " 56 G(α, β) #4%'%,%-' X "2 βα α−1 −βx x e Γ (α) f (x) = 0 x F(x) = α > x ≥ 0, α > 0, β > 0 β " ∞ Γ ( α ) = x α−1e − x dx 0 autrement f ( t ) dt −∞ " G ( α = 1, β ) ∞ " ; ∞ "" Γ (1) = e dx = − −e− x dx = − e− x −x 0 0 ∞ 0 f ( x ) = β e−βx . "' " & Γ (α) = β α ∞ x α−1e −βx dx 0 / Γ ( α ) = ( α − 1) Γ ( α − 1) α > 1 0 Γ ( n ) = ( n − 1) ! n ∈ N+ 1 Γ 1 = π 2 Γ (α) > = − [0 − 1] = 1 β α>0 " G(2,1) " A )"#$*'(&A/*$%*'(&&,4-'(,%-' 6#'#$*,$%(&0&) 5-5&',) α α β E ( X) = C V ( X) = 2 C Ψ ( t ) = β β β−t α 57 " 58 E(α, λ ) #4%'%,%-' " " α > E ( α, λ ) > β = λα % " α λ " X > " ( λα ) x α−1e−λαx Γ (α) α f (x) = 0 F(x) = x x ≥ 0, α > 0, λ > 0 autrement f ( t ) dt −∞ )"#$*'(&A/*$%*'(&&,4-'(,%-' 6#'#$*,$%(&0&) 5-5&',) 1 1 λα E ( X) = C V ( X) = C Ψ (t) = 2 λα − t λ αλ α G ( α, β ) " 2 α λ " 69 B(α, β) #4%'%,%-' " , X > Γ ( α + β ) α−1 β−1 x (1 − x ) f ( x ) = Γ ( α ) Γ (β) α β 0 ≤ x ≤ 1, α > 0, β > 0 0 F(x) = x f ( t ) dt −∞ B(2,2) " A )"#$*'(&&,/*$%*'(& E ( X) = α αβ C V ( X) = 2 α+β ( α + β ) ( α + β + 1) " " 6& #4%'%,%-' " f (x) = " 1 −x e 2 X " − ∞ < x < +∞ < $B$ 1 x e 2 f (x) = 1 −x e 2 x≤0 x≥0 1 x e 2 F(x) = 1 − e− x 1 1 + = 1 − e− x 2 2 2 x≤0 x≥0 " A )"#$*'(&A/*$%*'(&&,4-'(,%-' 6#'#$*,$%(&0&) 5-5&',) E ( X) = 0 C V ( X) = 2 C Ψ ( t ) = 1 1 1 + 2 1+ t 1− t " 6/ C χn2 DB @ #4%'%,%-' " α= n 2 f (x) = F(x) = χ32 B 0 E $ −∞ x n x −1 − 2 2 e x ≥ 0, n > 2 autrement f ( t ) dt " " A "2 " n 2 n Γ 2 0 x " 1 2 β= 1 2 ;Bn > " 60 )"#$*'(&A/*$%*'(&&,4-'(,%-' 6#'#$*,$%(&0&) 5-5&',) 1 E ( X ) = n C V ( X ) = 2n C Ψ ( t ) = 1 − 2t n 2 $-"$%#,# Xi ~ N ( 0, 1) Xi " Y= n i=1 Xi2 ~ χn2 " 61 tn #4%'%,%-' B n " n +1 2 f (x) = 1 n ( nπ ) 2 Γ 2 Γ F(x) = t5 B 4 x −∞ f ( t ) dt " " A 1+ 2 x n " − ( n+1) 2 − ∞ < x < +∞ " " )"#$*'(&&,/*$%*'(& E ( X) = 0 n > 1C V ( X ) = n n−2 n>2 $-"$%#,# X ~ N ( 0, 1) Y ~ χn2 " Z= X ~ tn Y n 64 " = B @ 65 Fm,n #4%'%,%-' " $ B m n " m n m+n m −1 Γ m 2 n2 x2 2 × m +n f (x) = m n Γ Γ (mx + n ) 2 2 2 0 autrement F(x) = F3,5 B 3 x −∞ f ( t )dt 5 " " A x≥0 " " 66 )"#$*'(&&,/*$%*'(& n E ( X) = n−2 n > 2 C V ( X) = 2n2 ( m + n − 2 ) m (n − 2 ) (n − 4 ) 2 $-"$%#,#) X ~ χm2 Y ~ χn2 " X Z = m ~ Fm,n Y n n>4