III
f x ( ) = 1 f x ( ) = P( X = x) = f ( x) = P( X = x) =! x ( ) b! a, X! U a,b f ( x) =! " e #!x, X! Ex (!) n! ( n! x)!x! " x 1! " x! e"!, X! Po! ( ) n! x, X! B( n;" ) ( )
! xf ( x) = = n n!! ( n " x)!x! x # $ x 1" $ x=0 n n!! $ ( n " x)! ( x 1" $ x " 1)! x=1 ( ) n" x n n! = $! $ ( n " x)! ( ( x"1) 1 " $ x " 1)! x=1 x-1=k n"1 n! =!#! ( k 1 "! n " k " 1)!k! k =0 ( ) n" x ( ) n" x n" k "1 ( ) m=n-1 m ( m + 1)! =!#! ( k 1 "! m " k)!k! k =0 ( ) =! m + 1 ( ) m" k m m! #! ( k 1 "! m " k)!k! k =0 { } m = n! = n!! + ( 1"!) ( ) m" k
! = x 2 f ( x) n n!! ( n " x)!x! x2 # $ x 1" $ x=0 ( ) n" x n n! = $! ( n " x)! ( x " 1)! x # $ x"1 1" $ x=1 ( ) n" x x-1=k m=n-1 m m! =! ( m + 1) $ ( ( m " k)!k! k " 1) #! k 1 "! k =0 m m! = n! $ ( ( m " k)!k! k " 1) #! k 1"! k =0 n % m! = n! & $ ( m " k)!k! k #! k 1"! ' k =1 ( ) m" k ( ) m" k + ( ) m" k n m! $ ( m " k)!k! #! k 1"! k =1 ( ) m" k ( ) *
m m! = n! $ ( m " k)!k! k #! k ( 1 "!) m" k + n! k =1 m ( m " 1)! = n! # m! $ ( m " k)! ( k " 1)! #! k "1 1"! k =1 ( ) m" k + n! "( x! x) 2 f ( x) = n# ( n! 1)# + n#! x 2 = n 2 # 2! n# 2 + n#! n 2 # 2 ( ) = n# 1! # k-1=i j=m-1 j j! = n! " m! $! ( i 1 #! j # i)!i! i=0 = n! " m! "(! + 1 #!) j + n! ( ) j #i + n! = n!m! + n! = n! ( n " 1)! + n! = n 2! 2 " n! 2 + n!
1! 1! 2
f ( x) = 1 2!" 2 e# ( x# µ ) 2 ( ) 2" 2, X! N µ," 2
X = ( X 1, X 2,...X n ) t ( ) = P( X 1! x 1,...X n! x n ) P X! x x n# =...!" x 1#!" f ( x 1,..., x n )dx 1...dx n
P( x! X! x + "x, y! Y! y + "y) x+ "x y+ "y ( ) = $ $ f x #, y # d y# d x# x y F( x, y) = P( X! x,y! y) x y ( ) = % % f x ", y " d y" d x" #$ #$
f 1 " " ( ) ( x) = #...# f x 1,..., x n dx 2...dx n!"!"
E!" g( X) #$ = g( x) f ( x)dx E g( X) % ( ( ) ( ) )!" #$ = E! " g x 1 #$,..., E! " g x n #$! µ = ( µ 1,..., µ n ) = E[ X] = ( E[ X 1 ],..., E[ X n ]) [ ] = x 1 f x E X 1! ( )dx =! x 1!...! f ( x)dx n...dx 1 =! x 1 f ( x 1 )dx 1
! i 2 = V X i [ ] = E #( X i " µ i ) 2! ij = Cov " # X i, X j $ # " 11! " 1n &! = V [ X] = % " # " ( % ( $ " n1! " nn ' % & ( ) X j & µ j $ % = E " # X & µ i i ( ) $ %
! ij = Corr " # X i, X j $ % = Cov " # X i, X j $ % V [ X i ]V " # X j $ %
P( ( X 1,..., X n ) = ( x 1..., x n )) = P( X 1 = x 1 )!...! P( X n = x n ) P( A! B) = P( A)P( B) ( ) = f 1 ( x 1 )!...! f n ( x n ) f x 1,..., x n F( x 1,..., x n ) = F 1 ( x 1 )!...! F n ( x n )
E[ X 1!...! X n ] = E[ X 1 ]!...! E[ X n ] E[ X 1! X 2 ] =!! x 1 x 2 f ( x 1, x 2 )dx 1 dx 2 =!! x 1 x 2 f 1 x 1 ( ) f 2 x 2 =! x 1 f 1 ( x 1 )dx 1 "! x 2 f 2 ( x 2 )dx 2 = E[ X 1 ]! E X 2 ( )dx 1 dx 2 [ ] [ ] = E ( X! µ X ) "( Y! µ Y ) Cov X,Y = 0 ( ) ( ) #$ % & = E "# X! µ X $ % & E "# Y! µ Y $ %
E[ X 1 + X 2 ]!! ( ) f x = x 1 + x 2 ( )dx 1 dx 2 =!! x 1 f ( x)dx 1 dx 2 +!! x 2 f ( x)dx 1 dx 2 = E[ X 1 ] + E[ X 2 ]
[ ] = E ( X 1 + X 2 )! ( µ 1 + µ 2 ) V X 1 + X 2 "( ) 2 # "( ) 2 ( ) + ( X 2! µ 2 ) = E X 1! µ # 1 ( ) 2 + 2( X 1! µ 1 ) "( X 2! µ 2 ) + ( X 2! µ 2 ) 2 = E # X 1! µ $ 1 = E "( X 1! µ 1 ) 2 $ # % + E "( X 2! µ 2 ) 2 # = V [ X 1 ] + V [ X 2 ] + 2Cov[ X 1, X 2 ] $ % $ % % & ( ) &( X 2! µ 2 ) $ % + 2E "# X 1! µ 1 $ %
( ) = P A! B P B A ( ) P( A) f Y X ( ) ( ) ( y x) = f x, y x f X
P X! µ " # ( ) $ % 2 # 2
Cov[ X,Y ] 2! V [ X]"V [ Y ] Cov[ X,Y ]! = V [ X]V [ Y ]