Mathematics 4 Statistics / 6. 89 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 (. http://wolfpack.hannam.ac.kr @5 Spring
Mathematics 4 Statistics / 6. 9. () 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. + 5 + c. -. C?.. C. d = + c] = ( + c) ( + c) = HOMEWORK #6- DUE 5 ( ). () = 6 + + ( ) = () = / (4) = 5 (5) = ( 6) = cos http://wolfpack.hannam.ac.kr @5 Spring
Mathematics 4 Statistics / 6. 9 6. ( ) ( - ). 7 Newton-Leiniz (integral).. [ a, ] f (? [ a, ].. ( : f ( ). 6.. Riemann [ a, ] f ( Riemann Integral( ). S r n = f ( ck ) k k= http://wolfpack.hannam.ac.kr @5 Spring
Mathematics 4 Statistics / 6. 9 Riemann. Riemann.?... 6... () 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) http://wolfpack.hannam.ac.kr @5 Spring
Mathematics 4 Statistics / 6. 9 6.. a Newton-Leiniz d = F( ) F( a). = 4 ) y = 4. ) y y = 4 (4 ) d = [4 ] = y = 4 - - ( 4) d = (4 ) d = y-.why?. y.. () [ a, ] =. () f ( ) = [ a, ] (su). ( y ). =. (). (4) http://wolfpack.hannam.ac.kr @5 Spring
Mathematics 4 Statistics / 6. 94 y = 4. ) y = ( 4) = ( )( + ) = =,, ) = [,] [,]. ). 4 ( 4 d = ] = 4 4 4 ( 4 d = ] = 4 4 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 http://wolfpack.hannam.ac.kr @5 Spring
Mathematics 4 Statistics / 6. 95 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 6 6 6 http://wolfpack.hannam.ac.kr @5 Spring
Mathematics 4 Statistics / 6. 96 + 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 = 7 + 5 du = d( 7 + 5) = 7d cos( 7 + 5) d cos u. 7 du cos u = cos udu = sin u + c = sin(7 + 5) + c 7 7 7 7 ( + ) ( + ) d. ( + ) ( + ) d = u du ( u = ( + ), du = ( + ) d ) = u + c 6 = ( + ) + c 6 http://wolfpack.hannam.ac.kr @5 Spring
Mathematics 4 Statistics / 6. 97 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 /. http://wolfpack.hannam.ac.kr @5 Spring
Mathematics 4 Statistics / 6. 98 6.4 () 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 http://wolfpack.hannam.ac.kr @5 Spring
Mathematics 4 Statistics / 6. 99 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 8 + 6.5 (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 ) http://wolfpack.hannam.ac.kr @5 Spring
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 http://wolfpack.hannam.ac.kr @5 Spring
Mathematics 4 Statistics / 6. sin d ( ) = f ( g( = sin g( + sin cos -- 6 + 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 = 6.6 6.6. (sample space) ( :element).? S = { : }, = http://wolfpack.hannam.ac.kr @5 Spring
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(. http://wolfpack.hannam.ac.kr @5 Spring
Mathematics 4 Statistics / 6. () P ( a X ) = a d () F( = P( X = d P( a < X ) = F( ) F( a). 6.6. ( ) (epected value). A={,,}, B={4,5} 5( ), C={6},5.,? (on average). 8. * + 5 * + 5 * = 8 6 6 6 X E ( X ) = p( X S E ( X ) = d. S http://wolfpack.hannam.ac.kr @5 Spring
Mathematics 4 Statistics / 6. 4 X.? E( X ) = P( = * + * + * + 4 * + 5* + 6* =.5 A 6 6 6 6 6 6 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 ] = =, 4 4 4 P ( Y ) = F() F() = ( ) ( ) = 4 4 4 http://wolfpack.hannam.ac.kr @5 Spring
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. 6.6. X (, ) ( f ( ) ( X ~ Ep( ) ). = e, <, < http://wolfpack.hannam.ac.kr @5 Spring
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 =. http://wolfpack.hannam.ac.kr @5 Spring
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. 6.6.4 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 http://wolfpack.hannam.ac.kr @5 Spring
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 http://wolfpack.hannam.ac.kr @5 Spring