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1 Mathematics 4 Statistics / Chapter 6 ( ), ( /) (Euclid geometry ( ), (( + )* /).? Archimedes,... (standard normal distriution, Gaussian distriution) X (..) (a, ). = ep{ } π σ a 6. f ( F ( = F( f ( - (anti-derivative)... f (, f ( - f ( (indefinite integral with respect to ) d. (integral), d f (. Spring

2 Mathematics 4 Statistics / () d = F( + c (c ) : () k d = k d ( ) () [ ± g( ] d = d ± g( d ( ) = d. d = + c. ( - ). 5 ( f =. 5 6 d = + c 6 ( + 5) d c. -. C?.. C. d = + c] = ( + c) ( + c) = HOMEWORK #6- DUE 5 ( ). () = ( ) = () = / (4) = 5 (5) = ( 6) = cos Spring

Mathematics 4 Statistics / 6. 9 6. ( ) ( - ).

3 Mathematics 4 Statistics / ( ) ( - ). 7 Newton-Leiniz (integral).. [ a, ] f (? [ a, ].. ( : f ( ). 6.. Riemann [ a, ] f ( Riemann Integral( ). S r n = f ( ck ) k k= Spring

4 Mathematics 4 Statistics / 6. 9 Riemann. Riemann.? () a a d = [ : ] a () a d = d ( ) () a k d = k a k d (k ) (4) a [ ± g( ] d = a d ± a g( d ( ) (5) [ a, ] g( a d a g( d (domination) (6) [ a, ] a d [ ] ( ). c c (7) a d + d = a d (8) a cd = c( a) Spring

5 Mathematics 4 Statistics / a Newton-Leiniz d = F( ) F( a). = 4 ) y = 4. ) y y = 4 (4 ) d = [4 ] = y = ( 4) d = (4 ) d = y-.why?. y.. () [ a, ] =. () f ( ) = [ a, ] (su). ( y ). =. (). (4) Spring

6 Mathematics 4 Statistics / y = 4. ) y = ( 4) = ( )( + ) = =,, ) = [,] [,]. ). 4 ( 4 d = ] = ( 4 d = ] = 4 4 4) = 8.?. ( ). HOMEWORK #6- DUE 5 ( ). + ( ) ( ) d d 5/ 6 ( ) (4 ) ( ) d ( + ) ( 4) d ( 5) π (sin ) d ( 6) d ) ( 7) ( s + ds ( 8) d 44 Spring

7 Mathematics 4 Statistics / HOMEWORK #6- DUE 5 ( ). 6.. n+ n u. u u du = + c. n + 5 ( + ) d. 5 u = + du = d( + ) = d ( + ) d u 5 du. 6 5 u ( + ) u du = + c = + c Spring

8 Mathematics 4 Statistics / d. = u = + du = d( + ) d + d u / du. / / / u ( + ) u du = + c = + c (sustitution). f ( g( ) g ( d = f ( u) du ( u = g(, du = g ( d ) = F ( u) + c = F ( g( ) + c () cos udu = sin u + c () sin udu = cos u + c () sec udu = tan u + c (4) csc udu = cot u + c (5) sec u tan udu = sec u + c (6) csc u cot udu = csc u + c cos( 7 + 5) d. du u = du = d( 7 + 5) = 7d cos( 7 + 5) d cos u. 7 du cos u = cos udu = sin u + c = sin(7 + 5) + c ( + ) ( + ) d. ( + ) ( + ) d = u du ( u = ( + ), du = ( + ) d ) = u + c 6 = ( + ) + c 6 Spring

9 Mathematics 4 Statistics / HOMEWORK #7- DUE 6 ( ). () sin( ) d ( ) 8(7 ) d ( ) ( 4 5 / dt ( 4) r (7 r ) dr ( 5) 8( ) d ( 6) dt 5t d HOMEWORK #7- DUE 6 ( ). 4 4 ( ) ( + ) d ( ) y y dy ( ) d + 4 dy (4) y ( + y ) ( 5) ( + d ) 5 ( 6) s + s (5s + ) ds 4 HOMEWORK #7- DUE 6 ( ). ) f d = f ( d. () d = d. a) f (even function) ) f (odd function) () a a h( d = a h( d h h. (4) h( = sin ( ), h( = cos ( ), a = π /. Spring

10 Mathematics 4 Statistics / () du = u + c () kdu = ku + c, k, ( ) () ( du ± dv) = du ± dv = u ± v ( ) n n+ (4) u du = u + c n + (5) n = du = ln u + c u ( ) ( ) cosudu = sin u + c ( ) sin udu = cosu + c ( ) tanudu = ln secu + c (6) ( ) e d = e + c ( ) a d = a + c ln a ( ) f e f ( ) = = e, = a f = a ln a (7) ( ) d = ln ( > ),ln( ( < ) ( ) du = tan u + c + u ( ) du = sin u + c u ( e + ) d. e d + d = e + ln + c Spring

11 Mathematics 4 Statistics / e ( ) d. u = ( ) u u du = d e d = e du = e + c = e + c d = ( + ) + du = tan u + c. + u d = du = tan u + c = tan ( + ) + c u HOMEWORK #7-4 DUE 6 ( ). ln ( ) d + ( ) d 6 ( ) d (integral y parts) d ( uv) = udv + vdu. d ( uv) = udv + vdu udv = uv vdu cos d. u =, dv = cos d. du = d, dv = cos d v = sin cos d = sin sin d = sin + cos + c ( udv = uv vdu ) Spring

12 Mathematics 4 Statistics / 6. ln d. e u = ln, dv = d. du = d (*), dv = d v = ln d = ln d = ln + c ( udv = uv vdu ) d. [ ] u =, dv = e d. du = d, dv = e d v = e e d = e e d ( udv = uv vdu ) u =, dv = e d. du = d, dv e d v = e = e d = e d = [ e e d] = ( e e ) ( udv = uv vdu ) e d = e e d = e e + e + c g( d f ( g( Taular integration( )., = f ( g ( = e g( + e -- e + e e e d = e e + e + c Spring

13 Mathematics 4 Statistics / 6. sin d ( ) = f ( g( = sin g( + sin cos sin 6 -- cos sin sin d = cos + sin + 6 cos 6 sin + c HOMEWORK #8- DUE 6 7 ( ). ( ) sin d = ( ) ln d = ( ) e d = ( 4) e d = (sample space) ( :element).? S = { : }, = Spring

14 Mathematics 4 Statistics / 6. (random variale) ( ) (real numer) ( ) ( ). S X. X ( s) =. X X. (, 58, 94, ) ( =, = 58, = 94,...). (finite), (infinite).,,,,.,. ( ), ( ) (density function), proanility density function) f ( ( p ( ).. () X. p ( () X. = p ( s s ( ) X f (. [ a, ] P( a X ) F(. Spring

Mathematics 4 Statistics / 6. () P ( a X ) = a d () F( = P( X = d P( a < X ) = F( ) F( a). 6.6. ( ) (epected value).

15 Mathematics 4 Statistics / 6. () P ( a X ) = a d () F( = P( X = d P( a < X ) = F( ) F( a) ( ) (epected value). A={,,}, B={4,5} 5( ), C={6},5.,? (on average). 8. * + 5 * + 5 * = X E ( X ) = p( X S E ( X ) = d. S Spring

16 Mathematics 4 Statistics / 6. 4 X.? E( X ) = P( = * + * + * + 4 * + 5* + 6* =.5 A X u( E ( u( X )) = u( P( ( E ( u( ) = u( d ). A u( X ) = ( X E( X )) (variance) (?).. : V ( X ) = E( X E( X )) = ( E( X )) p( : V ( X ) = E( X E( X )) = ( E( X )) d : V ( X ) = E( X ) E( X ) (..4. ) X () c. c(, =,. otherwise = c( >, c >. c( d = c ( )] [ ] = c = c = = /, () F (. = t t = F( dt t ] =, 4 4 () f ( F (. (4) P ( < Y ). t t P ( Y ) = dt = t ] = =, P ( Y ) = F() F() = ( ) ( ) = Spring

17 Mathematics 4 Statistics / 6. 5 (5) E(X ) V (X ). E ( X ) = d = ( ) d = [ ] = 6 E ( X ) = d = ( ) d = [ ] = 8 4 V ( X ) = E( X ) E( X ) = ( ) = 9 HOMEWORK #8- DUE 6 7 ( ) X () c. () F (. () P ( < X / ). (4) P ( X > / X >.). (5) E(X ) V (X ). c +, =,. otherwise HOMEWORK #8- DUE 6 7 ( ) X E (X ) = µ, V ( X ) = σ. Y = ax + ( a, ) E ( Y ) = aµ +, V ( X ) = a σ X (, ) ( f ( ) ( X ~ Ep( ) ). = e, <, < Spring

18 Mathematics 4 Statistics / 6. 6 = e. () < = e = >. () ( ) ] ( ) = = f d e d e = =. e X ~ Ep( ) E(X ) =. E( X ) = d = e d [ ] : udv = uv vdu u =, dv = e d du = d, dv e d v e = = E ( X ) = [ ] ] = = = e d ( e )] e d e d e = = f ( / g( = e g( / / e d = e e e + -- / e e / e / / / E ( X ) = ] d = [ e e = HOMEWORK #8-4 DUE 6 7 ( ) X ~ Ep( ) V (X ). V ( X ) = E( X ) E( X ). E(X ) = E( X ). E( X ) = d =. Spring

19 Mathematics 4 Statistics / 6. 7 HOMEWORK #8-5 DUE 6 7 ( ) (Gamma) Γ ) = n n y e dy = ( n ) Γ( n ). ( u = y n y, dv = e dy α y Γ( α) = y e dy = y d = dy, = y =, α / = y = Γ( α) = ( ) e ( ) d. α / α = = e d α Γ( α). α / = e,. α Γ( α) X ( α, ). X ~ Gamma( α, ), α = Gamma( α =, ) ( Ep ( ) ).. i ~ Ep( ) X ~ Gamma( α, ). ( Ep ( = ) ) ( α = ).. X X ~ Gamma( α =, = ). X iid α i= i Gamma α / α / ( α, β ) E( X ) = e d = e d α α Γ( α) Γ( α) α Γ( α) α / e d. α. α / α / / u =, dv = e du = α d, v = e d = βe Spring

20 Mathematics 4 Statistics / 6. 8 Γ( α) = Γ( α) α / α e d = [ + α α Γ( α) = α Γ( α) α / {[ ( e ) α α ( e α / e α α / α ) d] = Γ( α) ) d = αβ ( e / ) d]} α α α ( e / ) d α i ~ Ep( ) X ~ Gamma( α, ) X iid i= E (X ), V (X ). i X ~ Gamma( α, ) [ ] e d. u =, dv = e d. du = d, dv = e d v = e e d = e e d ( udv = uv vdu ) u =, dv = e d. du = d, dv = e d v = e e d = e d = [ e e d] = ( e e ) ( udv = uv vdu ) e d = e e d = e e + e + c ] g( d f ( g( Taular integration. =, g ( = e f ( g( e e e e e d = e e + e Spring

MS_적분.pages

MS_적분.pages 고대수학자들은사각형의면적 밑변 높이, 삼각형면적 밑변 높이 평행사변형의면적 Euclid gomtry 밑면 높이, 사다리꼴의면적 윗변 + 아래변 * 높이 를이용하여구하였다. 이를이용하여왼쪽의다각형면적은구할수있으나오른쪽의곡선의면적은어떻게구할것인가? Archimds 는곡선의면적을이미알려진다각형, 삼각형의면적으로근사시켜구하는방법을생각하였다. 이것이면적에대한현재정의의근간이된다.