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 ТерВер. Коэффициент корреляции
Сообщение09.12.2009, 21:33 


27/03/08
63
Здравствуйте!

Проверьте, пожалуйста, решение задачи.

Двухмерный случайный вектор (X,Y) равномерно распределён внутри выделенной жирными линиями области В.
Изображение

Двухмерная плотность вероятности $% MathType!MTEF!2!1!+-
% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9
% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x
% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacI
% cacaWG4bGaaiilaiaadMhacaGGPaaaaa!3ADC!
\[
{f(x,y)}
\] одинакова для любой точки этой области В.
$% MathType!MTEF!2!1!+-
% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9
% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x
% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacI
% cacaWG4bGaaiilaiaadMhacaGGPaGaeyypa0Zaaiqaaqaabeqaaiaa
% dogacaGG7aWaaeWaaeaacaWG4bGaaiilaiaadMhaaiaawIcacaGLPa
% aacqGHiiIZcaWGcbGaai4oaaqaaiaaicdacaGG7aGaamioeiaad2db
% caWGWqGaam4reiaadwdbaaGaay5Eaaaaaa!4B2F!
\[
f(x,y) = \left\{ \begin{array}{l}
 c;\left( {x,y} \right) \in B; \\ 
 0; \\ 
 \end{array} \right.
\]
$. Вычислить коэффициент корреляции между величинами X и Y.

Решение.

Исходя из рисунка, получаем:
$% MathType!MTEF!2!1!+-
% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9
% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x
% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacI
% cacaWG4bGaaiilaiaadMhacaGGPaGaeyypa0Zaaiqaaqaabeqaaiaa
% dogacaGG7aGaamyEaiabgsMiJkaadIhacqGHKjYOcaaI0aGaai4oai
% aaicdacqGHKjYOcaWG5bGaeyizImQaaGOmaiaacUdaaeaacaWGJbGa
% ai4oaiaaisdacqGH8aapcaWG4bGaeyizImQaaGOnaiaacUdacaaIWa
% GaeyizImQaamyEaiabgsMiJkaaigdacaGG7aaabaGaaGimaiaacUda
% caWG4qGaamypeiaadcdbcaWGhrGaamyneaaacaGL7baaaaa!5FA7!
\[
f(x,y) = \left\{ \begin{array}{l}
 c;y \le x \le 4;0 \le y \le 2; \\ 
 c;4 < x \le 6;0 \le y \le 1; \\ 
 0; \\ 
 \end{array} \right.
\]
$

Тогда
$% MathType!MTEF!2!1!+-
% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9
% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x
% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaadaWdXa
% qaamaapedabaGaamOzaiaacIcacaWG4bGaaiilaiaadMhacaGGPaGa
% eyyXICTaamizaiaadIhacqGHflY1caWGKbGaamyEaaWcbaGaeyOeI0
% IaeyOhIukabaGaeyOhIukaniabgUIiYdaaleaacqGHsislcqGHEisP
% aeaacqGHEisPa0Gaey4kIipakiabg2da9maapifabaGaam4yaaWcba
% GaamOqaaqab0Gaey4kIiVaey4kIipakiabgwSixlaadsgacaWG4bGa
% amizaiaadMhacqGH9aqpcaWGJbWaa8qCaeaacaWGKbGaamyEaaWcba
% GaaGimaaqaaiaaigdaa0Gaey4kIipakmaapehabaGaamizaiaadIha
% cqGHRaWkcaWGJbaaleaacaaI0aaabaGaaGOnaaqdcqGHRiI8aOWaa8
% qCaeaacaWGKbGaamyEaaWcbaGaaGimaaqaaiaaikdaa0Gaey4kIipa
% kmaapehabaGaamizaiaadIhaaSqaaiaadMhaaeaacaaI0aaaniabgU
% IiYdGccqGH9aqpcaWGJbGaeyyXIC9aa8qCaeaacaaIYaGaamizaiaa
% dMhaaSqaaiaaicdaaeaacaaIXaaaniabgUIiYdGccqGHRaWkcaWGJb
% Waa8qCaeaadaqadaqaaiaaisdacqGHsislcaWG5baacaGLOaGaayzk
% aaaaleaacaaIWaaabaGaaGOmaaqdcqGHRiI8aOGaamizaiaadMhacq
% GH9aqpcaaIYaGaam4yaiabgUcaRmaaeiaabaGaam4yaiabgwSixlaa
% isdacaWG5baacaGLiWoadaqhaaWcbaGaaGimaaqaaiaaikdaaaGccq
% GHsisldaWcaaqaaiaadogaaeaacaaIYaaaamaaeiaabaGaamyEamaa
% CaaaleqabaGaaGOmaaaaaOGaayjcSdWaa0baaSqaaiaaicdaaeaaca
% aIYaaaaOGaeyypa0dabaGaeyypa0JaaGOmaiaadogacqGHRaWkcaaI
% 0aGaam4yaiabgwSixlaaikdacqGHsisldaWcaaqaaiaadogaaeaaca
% aIYaaaaiabgwSixlaaisdacqGH9aqpcaaIYaGaam4yaiabgUcaRiaa
% iIdacaWGJbGaeyOeI0IaaGOmaiaadogacqGH9aqpcaaI4aGaam4yai
% aacUdaaaaa!BBAB!
\[
\begin{array}{l}
 \int_{ - \infty }^\infty  {\int_{ - \infty }^\infty  {f(x,y) \cdot dx \cdot dy} }  =  \cdot dxdy = c\int\limits_0^1 {dy} \int\limits_4^6 {dx + c} \int\limits_0^2 {dy} \int\limits_y^4 {dx}  = c \cdot \int\limits_0^1 {2dy}  + c\int\limits_0^2 {\left( {4 - y} \right)} dy = 2c + \left. {c \cdot 4y} \right|_0^2  - \frac{c}{2}\left. {y^2 } \right|_0^2  =  \\ 
  = 2c + 4c \cdot 2 - \frac{c}{2} \cdot 4 = 2c + 8c - 2c = 8c; \\ 
 \end{array}
\]
$

Значит
$% MathType!MTEF!2!1!+-
% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9
% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x
% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGioaiaado
% gacqGH9aqpcaaIXaGaai4oaiabgkDiElaadogacqGH9aqpdaWcaaqa
% aiaaigdaaeaacaaI4aaaaiaacUdaaaa!40AE!
\[
8c = 1; \Rightarrow c = \frac{1}{8};
\]
$

Получаем
$% MathType!MTEF!2!1!+-
% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9
% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x
% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacI
% cacaWG4bGaaiilaiaadMhacaGGPaGaeyypa0Zaaiqaaqaabeqaamaa
% laaabaGaaGymaaqaaiaaiIdaaaGaai4oaiaadMhacqGHKjYOcaWG4b
% GaeyizImQaaGinaiaacUdacaaIWaGaeyizImQaamyEaiabgsMiJkaa
% ikdacaGG7aaabaWaaSaaaeaacaaIXaaabaGaaGioaaaacaGG7aGaaG
% inaiabgYda8iaadIhacqGHKjYOcaaI2aGaai4oaiaaicdacqGHKjYO
% caWG5bGaeyizImQaaGymaiaacUdaaeaacaaIWaGaai4oaiaadIdbca
% WG9qGaamimeiaadEebcaWG1qaaaiaawUhaaaaa!60F1!
\[
f(x,y) = \left\{ \begin{array}{l}
 \frac{1}{8};y \le x \le 4;0 \le y \le 2; \\ 
 \frac{1}{8};4 < x \le 6;0 \le y \le 1; \\ 
 0; \\ 
 \end{array} \right.
\]
$

Вычисляем значения математических ожиданий
$% MathType!MTEF!2!1!+-
% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9
% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x
% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGTb
% WaaSbaaSqaaiaadIhaaeqaaOGaeyypa0Zaa8qmaeaadaWdXaqaaiaa
% dIhacqGHflY1caWGMbGaaiikaiaadIhacaGGSaGaamyEaiaacMcacq
% GHflY1caWGKbGaamiEaiabgwSixlaadsgacaWG5baaleaacqGHsisl
% cqGHEisPaeaacqGHEisPa0Gaey4kIipaaSqaaiabgkHiTiabg6HiLc
% qaaiabg6HiLcqdcqGHRiI8aOGaeyypa0ZaaSaaaeaacaaIXaaabaGa
% aGioaaaadaWdsbqaaiaadIhaaSqaaiaadkeaaeqaniabgUIiYlabgU
% IiYdGccqGHflY1caWGKbGaamiEaiaadsgacaWG5bGaeyypa0ZaaSaa
% aeaacaaIXaaabaGaaGioaaaadaWdXbqaaiaadsgacaWG5baaleaaca
% aIWaaabaGaaGymaaqdcqGHRiI8aOWaa8qCaeaacaWG4bGaamizaiaa
% dIhacqGHRaWkdaWcaaqaaiaaigdaaeaacaaI4aaaaaWcbaGaaGinaa
% qaaiaaiAdaa0Gaey4kIipakmaapehabaGaamizaiaadMhaaSqaaiaa
% icdaaeaacaaIYaaaniabgUIiYdGcdaWdXbqaaiaadIhacaWGKbGaam
% iEaaWcbaGaamyEaaqaaiaaisdaa0Gaey4kIipakiabg2da9maalaaa
% baGaaGymaaqaaiaaiIdaaaWaa8qCaeaadaqadaqaamaaeiaabaWaaS
% aaaeaacaWG4bWaaWbaaSqabeaacaaIYaaaaaGcbaGaaGOmaaaaaiaa
% wIa7amaaDaaaleaacaaIWaaabaGaaGOmaaaaaOGaayjkaiaawMcaai
% aadsgacaWG5baaleaacaaIWaaabaGaaGymaaqdcqGHRiI8aOGaey4k
% aSYaaSaaaeaacaaIXaaabaGaaGioaaaadaWdXbqaamaabmaabaWaaq
% GaaeaadaWcaaqaaiaadIhadaahaaWcbeqaaiaaikdaaaaakeaacaaI
% YaaaaaGaayjcSdWaa0baaSqaaiaadMhaaeaacaaI0aaaaaGccaGLOa
% GaayzkaaGaamizaiaadMhaaSqaaiaaicdaaeaacaaIYaaaniabgUIi
% YdGccqGH9aqpaeaacqGH9aqpdaWcaaqaaiaaigdaaeaacaaI4aaaam
% aapehabaWaaeWaaeaadaWcaaqaaiaaiodacaaI2aGaeyOeI0IaaGym
% aiaaiAdaaeaacaaIYaaaaaGaayjkaiaawMcaaaWcbaGaaGimaaqaai
% aaigdaa0Gaey4kIipakiaadsgacaWG5bGaey4kaSYaaSaaaeaacaaI
% XaaabaGaaGioaaaadaWdXbqaamaabmaabaWaaSaaaeaacaaIXaGaaG
% OnaiabgkHiTiaadMhadaahaaWcbeqaaiaaikdaaaaakeaacaaIYaaa
% aaGaayjkaiaawMcaaaWcbaGaaGimaaqaaiaaikdaa0Gaey4kIipaki
% aadsgacaWG5bGaeyypa0ZaaSaaaeaacaaIYaGaaGinaaqaaiaaiIda
% aaGaeyypa0JaaG4maiaacUdaaaaa!C71F!
\[
\begin{array}{l}
 m_x  = \int_{ - \infty }^\infty  {\int_{ - \infty }^\infty  {x \cdot f(x,y) \cdot dx \cdot dy} }  = \frac{1}{8} \cdot dxdy = \frac{1}{8}\int\limits_0^1 {dy} \int\limits_4^6 {xdx + \frac{1}{8}} \int\limits_0^2 {dy} \int\limits_y^4 {xdx}  = \frac{1}{8}\int\limits_0^1 {\left( {\left. {\frac{{x^2 }}{2}} \right|_0^2 } \right)dy}  + \frac{1}{8}\int\limits_0^2 {\left( {\left. {\frac{{x^2 }}{2}} \right|_y^4 } \right)dy}  =  \\ 
  = \frac{1}{8}\int\limits_0^1 {\left( {\frac{{36 - 16}}{2}} \right)} dy + \frac{1}{8}\int\limits_0^2 {\left( {\frac{{16 - y^2 }}{2}} \right)} dy = \frac{{24}}{8} = 3; \\ 
 \end{array}
\]
$


$% MathType!MTEF!2!1!+-
% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9
% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x
% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGTb
% WaaSbaaSqaaiaadMhaaeqaaOGaeyypa0Zaa8qmaeaadaWdXaqaaiaa
% dMhacqGHflY1caWGMbGaaiikaiaadIhacaGGSaGaamyEaiaacMcacq
% GHflY1caWGKbGaamiEaiabgwSixlaadsgacaWG5baaleaacqGHsisl
% cqGHEisPaeaacqGHEisPa0Gaey4kIipaaSqaaiabgkHiTiabg6HiLc
% qaaiabg6HiLcqdcqGHRiI8aOGaeyypa0ZaaSaaaeaacaaIXaaabaGa
% aGioaaaadaWdsbqaaiaadMhaaSqaaiaadkeaaeqaniabgUIiYlabgU
% IiYdGccqGHflY1caWGKbGaamiEaiaadsgacaWG5bGaeyypa0ZaaSaa
% aeaacaaIXaaabaGaaGioaaaadaWdXbqaaiaadMhacaWGKbGaamyEaa
% WcbaGaaGimaaqaaiaaigdaa0Gaey4kIipakmaapehabaGaamizaiaa
% dIhacqGHRaWkdaWcaaqaaiaaigdaaeaacaaI4aaaaaWcbaGaaGinaa
% qaaiaaiAdaa0Gaey4kIipakmaapehabaGaamyEaiaadsgacaWG5baa
% leaacaaIWaaabaGaaGOmaaqdcqGHRiI8aOWaa8qCaeaacaWGKbGaam
% iEaaWcbaGaamyEaaqaaiaaisdaa0Gaey4kIipakiabg2da9maalaaa
% baGaaGymaaqaaiaaiIdaaaWaa8qCaeaacaaIYaGaamyEaiaadsgaca
% WG4bGaey4kaSYaaSaaaeaacaaIXaaabaGaaGioaaaaaSqaaiaaicda
% aeaacaaIXaaaniabgUIiYdGcdaWdXbqaaiaadMhadaqadaqaaiaais
% dacqGHsislcaWG5baacaGLOaGaayzkaaGaamizaiaadMhaaSqaaiaa
% icdaaeaacaaIYaaaniabgUIiYdGccqGH9aqpaeaacqGH9aqpdaWcaa
% qaaiaaiodacaaI1aaabaGaaGOmaiaaisdaaaGaeyisISRaaGymaiaa
% c6cacaaI0aGaaGOnaiaacUdaaaaa!A4C1!
\[
\begin{array}{l}
 m_y  = \int_{ - \infty }^\infty  {\int_{ - \infty }^\infty  {y \cdot f(x,y) \cdot dx \cdot dy} }  = \frac{1}{8} \cdot dxdy = \frac{1}{8}\int\limits_0^1 {ydy} \int\limits_4^6 {dx + \frac{1}{8}} \int\limits_0^2 {ydy} \int\limits_y^4 {dx}  = \frac{1}{8}\int\limits_0^1 {2ydx + \frac{1}{8}} \int\limits_0^2 {y\left( {4 - y} \right)dy}  =  \\ 
  = \frac{{35}}{{24}} \approx 1.46; \\ 
 \end{array}
\]
$


Далее, вычисляем дисперсии
$% MathType!MTEF!2!1!+-
% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9
% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x
% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGeb
% WaaSbaaSqaaiaadIhaaeqaaOGaeyypa0Zaa8qmaeaadaWdXaqaaiaa
% dIhadaahaaWcbeqaaiaaikdaaaGccqGHflY1caWGMbGaaiikaiaadI
% hacaGGSaGaamyEaiaacMcacqGHflY1caWGKbGaamiEaiabgwSixlaa
% dsgacaWG5baaleaacqGHsislcqGHEisPaeaacqGHEisPa0Gaey4kIi
% paaSqaaiabgkHiTiabg6HiLcqaaiabg6HiLcqdcqGHRiI8aOGaeyOe
% I0IaamyBamaaBaaaleaacaWG4baabeaakmaaCaaaleqabaGaaGOmaa
% aakiabg2da9maalaaabaGaaGymaaqaaiaaiIdaaaWaa8qCaeaacaWG
% KbGaamyEaaWcbaGaaGimaaqaaiaaigdaa0Gaey4kIipakmaapehaba
% GaamiEamaaCaaaleqabaGaaGOmaaaakiaadsgacaWG4bGaey4kaSYa
% aSaaaeaacaaIXaaabaGaaGioaaaaaSqaaiaaisdaaeaacaaI2aaani
% abgUIiYdGcdaWdXbqaaiaadsgacaWG5baaleaacaaIWaaabaGaaGOm
% aaqdcqGHRiI8aOWaa8qCaeaacaWG4bWaaWbaaSqabeaacaaIYaaaaO
% GaamizaiaadIhaaSqaaiaadMhaaeaacaaI0aaaniabgUIiYdGccqGH
% sisldaqadaqaaiaaiodaaiaawIcacaGLPaaadaahaaWcbeqaaiaaik
% daaaGccqGH9aqpdaWcaaqaaiaaigdaaeaacaaI4aaaamaapehabaWa
% aeWaaeaadaabcaqaamaalaaabaGaamiEamaaCaaaleqabaGaaG4maa
% aaaOqaaiaaiodaaaaacaGLiWoadaqhaaWcbaGaaGinaaqaaiaaiAda
% aaaakiaawIcacaGLPaaacaWGKbGaamyEaaWcbaGaaGimaaqaaiaaig
% daa0Gaey4kIipakiabgUcaRmaalaaabaGaaGymaaqaaiaaiIdaaaWa
% a8qCaeaadaqadaqaamaaeiaabaWaaSaaaeaacaWG4bWaaWbaaSqabe
% aacaaIZaaaaaGcbaGaaG4maaaaaiaawIa7amaaDaaaleaacaWG5baa
% baGaaGinaaaaaOGaayjkaiaawMcaaiaadsgacaWG5bGaeyOeI0Yaae
% WaaeaacaaIZaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaaqa
% aiaaicdaaeaacaaIYaaaniabgUIiYdGccqGH9aqpaeaacqGH9aqpca
% aIXaGaaGymaiaac6cacaaI1aGaeyOeI0IaaGyoaiabg2da9iaaikda
% caGGUaGaaGynaiaacUdaaaaa!ADBB!
\[
\begin{array}{l}
 D_x  = \int_{ - \infty }^\infty  {\int_{ - \infty }^\infty  {x^2  \cdot f(x,y) \cdot dx \cdot dy} }  - m_x ^2  = \frac{1}{8}\int\limits_0^1 {dy} \int\limits_4^6 {x^2 dx + \frac{1}{8}} \int\limits_0^2 {dy} \int\limits_y^4 {x^2 dx}  - \left( 3 \right)^2  = \frac{1}{8}\int\limits_0^1 {\left( {\left. {\frac{{x^3 }}{3}} \right|_4^6 } \right)dy}  + \frac{1}{8}\int\limits_0^2 {\left( {\left. {\frac{{x^3 }}{3}} \right|_y^4 } \right)dy - \left( 3 \right)^2 }  =  \\ 
  = 11.5 - 9 = 2.5; \\ 
 \end{array}
\]
$
$% MathType!MTEF!2!1!+-
% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9
% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x
% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGeb
% WaaSbaaSqaaiaadMhaaeqaaOGaeyypa0Zaa8qmaeaadaWdXaqaaiaa
% dMhadaahaaWcbeqaaiaaikdaaaGccqGHflY1caWGMbGaaiikaiaadI
% hacaGGSaGaamyEaiaacMcacqGHflY1caWGKbGaamiEaiabgwSixlaa
% dsgacaWG5baaleaacqGHsislcqGHEisPaeaacqGHEisPa0Gaey4kIi
% paaSqaaiabgkHiTiabg6HiLcqaaiabg6HiLcqdcqGHRiI8aOGaeyOe
% I0IaamyBamaaBaaaleaacaWG5baabeaakmaaCaaaleqabaGaaGOmaa
% aakiabg2da9maalaaabaGaaGymaaqaaiaaiIdaaaWaa8qCaeaacaWG
% 5bWaaWbaaSqabeaacaaIYaaaaOGaamizaiaadMhaaSqaaiaaicdaae
% aacaaIXaaaniabgUIiYdGcdaWdXbqaaiaadsgacaWG4bGaey4kaSYa
% aSaaaeaacaaIXaaabaGaaGioaaaaaSqaaiaaisdaaeaacaaI2aaani
% abgUIiYdGcdaWdXbqaaiaadMhadaahaaWcbeqaaiaaikdaaaGccaWG
% KbGaamyEaaWcbaGaaGimaaqaaiaaikdaa0Gaey4kIipakmaapehaba
% GaamizaiaadIhaaSqaaiaadMhaaeaacaaI0aaaniabgUIiYdGccqGH
% sisldaqadaqaaiaaigdacaGGUaGaaGinaiaaiAdaaiaawIcacaGLPa
% aadaahaaWcbeqaaiaaikdaaaGccqGH9aqpdaWcaaqaaiaaigdaaeaa
% caaI4aaaamaapehabaGaaGOmaiaadMhadaahaaWcbeqaaiaaikdaaa
% GccaWGKbGaamyEaiabgUcaRmaalaaabaGaaGymaaqaaiaaiIdaaaaa
% leaacaaIWaaabaGaaGymaaqdcqGHRiI8aOWaa8qCaeaacaWG5bWaaW
% baaSqabeaacaaIYaaaaOWaaeWaaeaacaaI0aGaeyOeI0IaamyEaaGa
% ayjkaiaawMcaaiaadsgacaWG5baaleaacaaIWaaabaGaaGOmaaqdcq
% GHRiI8aOGaeyOeI0IaaGOmaiaac6cacaaIXaGaaG4maiabg2da9aqa
% aiabg2da9maalaaabaGaaGymaiaaigdaaeaacaaIXaGaaGOmaaaacq
% GHijYUcaaIWaGaaiOlaiaaiMdacaaIXaGaai4oaaaaaa!A8EC!
\[
\begin{array}{l}
 D_y  = \int_{ - \infty }^\infty  {\int_{ - \infty }^\infty  {y^2  \cdot f(x,y) \cdot dx \cdot dy} }  - m_y ^2  = \frac{1}{8}\int\limits_0^1 {y^2 dy} \int\limits_4^6 {dx + \frac{1}{8}} \int\limits_0^2 {y^2 dy} \int\limits_y^4 {dx}  - \left( {1.46} \right)^2  = \frac{1}{8}\int\limits_0^1 {2y^2 dy + \frac{1}{8}} \int\limits_0^2 {y^2 \left( {4 - y} \right)dy}  - 2.13 =  \\ 
  = \frac{{11}}{{12}} \approx 0.91; \\ 
 \end{array}
\]
$

Вычисляем значения среднеквадратичных отклонений величин X и Y:
$% MathType!MTEF!2!1!+-
% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9
% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x
% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS
% baaSqaaiaadIhaaeqaaOGaeyypa0ZaaOaaaeaacaWGebWaaSbaaSqa
% aiaadIhaaeqaaaqabaGccqGH9aqpdaGcaaqaaiaaikdacaGGUaGaaG
% ynaaWcbeaakiabgIKi7kaaigdacaGGUaGaaGynaiaaiIdacaGG7aaa
% aa!44AB!
\[
\sigma _x  = \sqrt {D_x }  = \sqrt {2.5}  \approx 1.58;
\]
$


$% MathType!MTEF!2!1!+-
% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9
% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x
% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS
% baaSqaaiaadMhaaeqaaOGaeyypa0ZaaOaaaeaacaWGebWaaSbaaSqa
% aiaadMhaaeqaaaqabaGccqGH9aqpdaGcaaqaaiaaicdacaGGUaGaaG
% yoaiaaigdaaSqabaGccqGHijYUcaaIWaGaaiOlaiaaiMdacaaI1aGa
% ai4oaaaa!456A!
\[
\sigma _y  = \sqrt {D_y }  = \sqrt {0.91}  \approx 0.95;
\]
$

Корреляционный момент
$% MathType!MTEF!2!1!+-
% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9
% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x
% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGlb
% WaaSbaaSqaaiaadIhacaWG5baabeaakiabg2da9maapedabaWaa8qm
% aeaacaWG4bGaeyyXICTaamyEaiabgwSixlaadAgacaGGOaGaamiEai
% aacYcacaWG5bGaaiykaiabgwSixlaadsgacaWG4bGaeyyXICTaamiz
% aiaadMhaaSqaaiabgkHiTiabg6HiLcqaaiabg6HiLcqdcqGHRiI8aa
% WcbaGaeyOeI0IaeyOhIukabaGaeyOhIukaniabgUIiYdGccqGHsisl
% daqadaqaaiaad2gadaWgaaWcbaGaamiEaaqabaGccqGHflY1caWGTb
% WaaSbaaSqaaiaadIhaaeqaaaGccaGLOaGaayzkaaGaeyypa0ZaaSaa
% aeaacaaIXaaabaGaaGioaaaadaWdXbqaaiaadMhacaWGKbGaamyEaa
% WcbaGaaGimaaqaaiaaigdaa0Gaey4kIipakmaapehabaGaamiEaiaa
% dsgacaWG4bGaey4kaSYaaSaaaeaacaaIXaaabaGaaGioaaaaaSqaai
% aaisdaaeaacaaI2aaaniabgUIiYdGcdaWdXbqaaiaadMhacaWGKbGa
% amyEaaWcbaGaaGimaaqaaiaaikdaa0Gaey4kIipakmaapehabaGaam
% iEaiaadsgacaWG4baaleaacaWG5baabaGaaGinaaqdcqGHRiI8aOGa
% eyOeI0IaaGinaiaac6cacaaIZaGaaGioaiabg2da9maalaaabaGaaG
% ymaaqaaiaaiIdaaaWaa8qCaeaadaqadaqaamaaeiaabaWaaSaaaeaa
% caWG4bWaaWbaaSqabeaacaaIYaaaaaGcbaGaaGOmaaaaaiaawIa7am
% aaDaaaleaacaaI0aaabaGaaGOnaaaaaOGaayjkaiaawMcaaiaadMha
% caWGKbGaamyEaaWcbaGaaGimaaqaaiaaigdaa0Gaey4kIipakiabgU
% caRmaalaaabaGaaGymaaqaaiaaiIdaaaWaa8qCaeaadaqadaqaamaa
% eiaabaWaaSaaaeaacaWG4bWaaWbaaSqabeaacaaIYaaaaaGcbaGaaG
% OmaaaaaiaawIa7amaaDaaaleaacaWG5baabaGaaGinaaaaaOGaayjk
% aiaawMcaaiaadMhacaWGKbGaamyEaaWcbaGaaGimaaqaaiaaikdaa0
% Gaey4kIipakiabgkHiTiaaisdacaGGUaGaaG4maiaaiIdacqGH9aqp
% aeaacqGH9aqpcqGHsislcaaIYaGaaiOlaiaaicdacaaIWaGaaGynai
% aacUdaaaaa!B46D!
\[
\begin{array}{l}
 K_{xy}  = \int_{ - \infty }^\infty  {\int_{ - \infty }^\infty  {x \cdot y \cdot f(x,y) \cdot dx \cdot dy} }  - \left( {m_x  \cdot m_x } \right) = \frac{1}{8}\int\limits_0^1 {ydy} \int\limits_4^6 {xdx + \frac{1}{8}} \int\limits_0^2 {ydy} \int\limits_y^4 {xdx}  - 4.38 = \frac{1}{8}\int\limits_0^1 {\left( {\left. {\frac{{x^2 }}{2}} \right|_4^6 } \right)ydy}  + \frac{1}{8}\int\limits_0^2 {\left( {\left. {\frac{{x^2 }}{2}} \right|_y^4 } \right)ydy}  - 4.38 =  \\ 
  =  - 2.005; \\ 
 \end{array}
\]
$

Коэффициент корреляции
$% MathType!MTEF!2!1!+-
% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9
% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x
% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyWdi3aaS
% baaSqaaiaadIhacaWG5baabeaakiabg2da9maalaaabaGaam4samaa
% BaaaleaacaWG4bGaamyEaaqabaaakeaacqaHdpWCdaWgaaWcbaGaam
% yEaaqabaGccqaHdpWCdaWgaaWcbaGaamiEaaqabaaaaOGaeyypa0Za
% aSaaaeaacqGHsislcaaIYaGaaiOlaiaaicdacaaIWaGaaGynaaqaai
% aaigdacaGGUaGaaGynaiaaiIdacqGHflY1caaIWaGaaiOlaiaaiMda
% caaI1aaaaiabg2da9iabgkHiTiaaigdacaGGUaGaaG4maiaaiodaca
% GG7aaaaa!5745!
\[
\rho _{xy}  = \frac{{K_{xy} }}{{\sigma _y \sigma _x }} = \frac{{ - 2.005}}{{1.58 \cdot 0.95}} =  - 1.33;
\]
$

 Профиль  
                  
 
 Re: ТерВер. Коэффициент корреляции
Сообщение09.12.2009, 21:47 
Заслуженный участник
Аватара пользователя


18/05/06
13438
с Территории
Ничего не читал, кроме последней строчки, но такие коэффициенты корреляции бывают только в военное время.

 Профиль  
                  
 
 Re: ТерВер. Коэффициент корреляции
Сообщение09.12.2009, 21:50 
Заслуженный участник
Аватара пользователя


11/04/08
2748
Физтех
Ошибки в обоих матожиданиях.

 Профиль  
                  
 
 Re: ТерВер. Коэффициент корреляции
Сообщение09.12.2009, 22:54 


27/03/08
63
Спасибо за ответы. Я подозревал, что допущены ошибки в вычислениях.
Меня интересует следующий момент: правильно ли выбраны пределы интегрирования? Т.е., по сути, правильно ли я определил область B? Просто у меня такое чувство, что ошибки в вычислениях обусловлены именно неправильно выбранными пределами интегрирования.

 Профиль  
                  
 
 Re: ТерВер. Коэффициент корреляции
Сообщение09.12.2009, 23:28 
Супермодератор
Аватара пользователя


29/07/05
8248
Москва
Первая часть графика дает ситуацию, когда в повторном интеграле пределы изменения внутреннего не постоянны, а зависят от внешнего.

Интеграл чего угодно по этой области будет выглядеть так:

$\int_0^2\,dx\int_0^x\,dy + \int_2^4\,dx\int_0^2\,dy + \int_4^6\,dx\int_0^1\,dy$

 Профиль  
                  
 
 Re: ТерВер. Коэффициент корреляции
Сообщение10.12.2009, 19:21 
Заслуженный участник
Аватара пользователя


23/11/06
4171
PAV в сообщении #269629 писал(а):
Первая часть графика дает ситуацию, когда в повторном интеграле пределы изменения внутреннего не постоянны, а зависят от внешнего.

Интеграл чего угодно по этой области будет выглядеть так:

$\int_0^2\,dx\int_0^x\,dy + \int_2^4\,dx\int_0^2\,dy + \int_4^6\,dx\int_0^1\,dy$

Вроде бы это ровно то же самое, что интегралы у автора: $\int_0^2\,dy \int_y^4\,dx + \int_0^1\,dy \int_4^6\,dx$ :)

 Профиль  
                  
 
 Re: ТерВер. Коэффициент корреляции
Сообщение10.12.2009, 23:43 


27/03/08
63
В общем, ещё раз внимательно проверив рассчёты получил следующее.

(Условие и рисунок тот же)
$% MathType!MTEF!2!1!+-
% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9
% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x
% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaadaWdXa
% qaamaapedabaGaamOzaiaacIcacaWG4bGaaiilaiaadMhacaGGPaGa
% eyyXICTaamizaiaadIhacqGHflY1caWGKbGaamyEaaWcbaGaeyOeI0
% IaeyOhIukabaGaeyOhIukaniabgUIiYdaaleaacqGHsislcqGHEisP
% aeaacqGHEisPa0Gaey4kIipakiabg2da9iaadogadaWadaqaamaape
% habaGaamizaiaadIhaaSqaaiaaicdaaeaacaaIYaaaniabgUIiYdGc
% daWdXbqaaiaadsgacaWG5bGaey4kaScaleaacaaIWaaabaGaamiEaa
% qdcqGHRiI8aOWaa8qCaeaacaWGKbGaamiEaaWcbaGaaGOmaaqaaiaa
% isdaa0Gaey4kIipakmaapehabaGaamizaiaadMhacqGHRaWkdaWdXb
% qaaiaadsgacaWG4baaleaacaaI0aaabaGaaGOnaaqdcqGHRiI8aOWa
% a8qCaeaacaWGKbGaamyEaaWcbaGaaGimaaqaaiaaigdaa0Gaey4kIi
% paaSqaaiaaicdaaeaacaaIYaaaniabgUIiYdaakiaawUfacaGLDbaa
% cqGH9aqpcaWGJbWaamWaaeaadaWdXbqaaiaadIhacaWGKbGaamiEai
% abgUcaRaWcbaGaaGimaaqaaiaaikdaa0Gaey4kIipakmaapehabaGa
% aGOmaiaadsgacaWG4baaleaacaaIYaaabaGaaGinaaqdcqGHRiI8aO
% Gaey4kaSYaa8qCaeaacaWGKbGaamiEaaWcbaGaaGinaaqaaiaaiAda
% a0Gaey4kIipaaOGaay5waiaaw2faaiabg2da9iaadogadaWadaqaam
% aalaaabaGaaGinaiabgkHiTiaaicdaaeaacaaIYaaaaiabgUcaRiaa
% ikdadaqadaqaaiaaisdacqGHsislcaaIYaaacaGLOaGaayzkaaGaey
% 4kaSYaaeWaaeaacaaI2aGaeyOeI0IaaGinaaGaayjkaiaawMcaaaGa
% ay5waiaaw2faaiabg2da9aqaaiabg2da9iaaiIdacaWGJbGaai4oaa
% aaaa!A6C1!
\[
\begin{array}{l}
 \int_{ - \infty }^\infty  {\int_{ - \infty }^\infty  {f(x,y) \cdot dx \cdot dy} }  = c\left[ {\int\limits_0^2 {dx} \int\limits_0^x {dy + } \int\limits_2^4 {dx} \int\limits_0^2 {dy + \int\limits_4^6 {dx} \int\limits_0^1 {dy} } } \right] = c\left[ {\int\limits_0^2 {xdx + } \int\limits_2^4 {2dx}  + \int\limits_4^6 {dx} } \right] = c\left[ {\frac{{4 - 0}}{2} + 2\left( {4 - 2} \right) + \left( {6 - 4} \right)} \right] =  \\ 
  = 8c; \\ 
 \end{array}
\]
$

Из условия нормировки получаем
$% MathType!MTEF!2!1!+-
% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9
% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x
% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGioaiaado
% gacqGH9aqpcaaIXaGaai4oaiabgkDiElaadogacqGH9aqpdaWcaaqa
% aiaaigdaaeaacaaI4aaaaiaacUdaaaa!40AE!
\[
8c = 1; \Rightarrow c = \frac{1}{8};
\]
$

Математические ожидания
$% MathType!MTEF!2!1!+-
% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9
% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x
% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaaBa
% aaleaacaWG4baabeaakiabg2da9maapedabaWaa8qmaeaacaWG4bGa
% eyyXICTaamOzaiaacIcacaWG4bGaaiilaiaadMhacaGGPaGaeyyXIC
% TaamizaiaadIhacqGHflY1caWGKbGaamyEaaWcbaGaeyOeI0IaeyOh
% IukabaGaeyOhIukaniabgUIiYdaaleaacqGHsislcqGHEisPaeaacq
% GHEisPa0Gaey4kIipakiabg2da9maalaaabaGaaGymaaqaaiaaiIda
% aaWaamWaaeaadaWdXbqaaiaadIhacaWGKbGaamiEaaWcbaGaaGimaa
% qaaiaaikdaa0Gaey4kIipakmaapehabaGaamizaiaadMhacqGHRaWk
% aSqaaiaaicdaaeaacaWG4baaniabgUIiYdGcdaWdXbqaaiaadIhaca
% WGKbGaamiEaaWcbaGaaGOmaaqaaiaaisdaa0Gaey4kIipakmaapeha
% baGaamizaiaadMhacqGHRaWkdaWdXbqaaiaadIhacaWGKbGaamiEaa
% WcbaGaaGinaaqaaiaaiAdaa0Gaey4kIipakmaapehabaGaamizaiaa
% dMhaaSqaaiaaicdaaeaacaaIXaaaniabgUIiYdaaleaacaaIWaaaba
% GaaGOmaaqdcqGHRiI8aaGccaGLBbGaayzxaaGaeyypa0ZaaSaaaeaa
% caaIXaaabaGaaGioaaaadaWadaqaamaapehabaGaamiEamaaCaaale
% qabaGaaGOmaaaakiaadsgacaWG4baaleaacaaIWaaabaGaaGOmaaqd
% cqGHRiI8aOGaey4kaSIaaGOmamaapehabaGaamiEaiaadsgacaWG4b
% aaleaacaaIYaaabaGaaGinaaqdcqGHRiI8aOGaey4kaSYaa8qCaeaa
% caWG4bGaamizaiaadIhaaSqaaiaaisdaaeaacaaI2aaaniabgUIiYd
% aakiaawUfacaGLDbaacqGH9aqpdaWcaaqaaiaaigdaaeaacaaI4aaa
% amaadmaabaWaaSaaaeaacaaI4aaabaGaaG4maaaacqGHRaWkdaqada
% qaaiaaigdacaaI2aGaeyOeI0IaaGinaaGaayjkaiaawMcaaiabgUca
% RmaabmaabaWaaSaaaeaacaaIZaGaaGOnaiabgkHiTiaaigdacaaI2a
% aabaGaaGOmaaaaaiaawIcacaGLPaaaaiaawUfacaGLDbaacqGH9aqp
% daWcaaqaaiaaiodacaaI3aaabaGaaGymaiaaikdaaaGaeyisISRaaG
% 4maiaac6cacaaIWaGaaGioaiaacUdaaaa!BA97!
\[
m_x  = \int_{ - \infty }^\infty  {\int_{ - \infty }^\infty  {x \cdot f(x,y) \cdot dx \cdot dy} }  = \frac{1}{8}\left[ {\int\limits_0^2 {xdx} \int\limits_0^x {dy + } \int\limits_2^4 {xdx} \int\limits_0^2 {dy + \int\limits_4^6 {xdx} \int\limits_0^1 {dy} } } \right] = \frac{1}{8}\left[ {\int\limits_0^2 {x^2 dx}  + 2\int\limits_2^4 {xdx}  + \int\limits_4^6 {xdx} } \right] = \frac{1}{8}\left[ {\frac{8}{3} + \left( {16 - 4} \right) + \left( {\frac{{36 - 16}}{2}} \right)} \right] = \frac{{37}}{{12}} \approx 3.08;
\]
$

$% MathType!MTEF!2!1!+-
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% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9
% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x
% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaaBa
% aaleaacaWG5baabeaakiabg2da9maapedabaWaa8qmaeaacaWG5bGa
% eyyXICTaamOzaiaacIcacaWG4bGaaiilaiaadMhacaGGPaGaeyyXIC
% TaamizaiaadIhacqGHflY1caWGKbGaamyEaaWcbaGaeyOeI0IaeyOh
% IukabaGaeyOhIukaniabgUIiYdaaleaacqGHsislcqGHEisPaeaacq
% GHEisPa0Gaey4kIipakiabg2da9maalaaabaGaaGymaaqaaiaaiIda
% aaWaamWaaeaadaWdXbqaaiaadsgacaWG4baaleaacaaIWaaabaGaaG
% OmaaqdcqGHRiI8aOWaa8qCaeaacaWG5bGaamizaiaadMhacqGHRaWk
% aSqaaiaaicdaaeaacaWG4baaniabgUIiYdGcdaWdXbqaaiaadsgaca
% WG4baaleaacaaIYaaabaGaaGinaaqdcqGHRiI8aOWaa8qCaeaacaWG
% 5bGaamizaiaadMhacqGHRaWkdaWdXbqaaiaadsgacaWG4baaleaaca
% aI0aaabaGaaGOnaaqdcqGHRiI8aOWaa8qCaeaacaWG5bGaamizaiaa
% dMhaaSqaaiaaicdaaeaacaaIXaaaniabgUIiYdaaleaacaaIWaaaba
% GaaGOmaaqdcqGHRiI8aaGccaGLBbGaayzxaaGaeyypa0ZaaSaaaeaa
% caaIXaaabaGaaGioaaaadaWadaqaamaalaaabaGaaGymaaqaaiaaik
% daaaWaa8qCaeaacaWG4bWaaWbaaSqabeaacaaIYaaaaOGaamizaiaa
% dIhaaSqaaiaaicdaaeaacaaIYaaaniabgUIiYdGccqGHRaWkdaWcaa
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% aIXaaabaGaaGOmaaaadaWdXbqaaiaaigdacaWGKbGaamiEaaWcbaGa
% aGinaaqaaiaaiAdaa0Gaey4kIipaaOGaay5waiaaw2faaiabg2da9m
% aalaaabaGaaGymaaqaaiaaiIdaaaWaamWaaeaadaWcaaqaaiaaisda
% aeaacaaIZaaaaiabgUcaRiaaisdacqGHRaWkcaaIXaaacaGLBbGaay
% zxaaGaeyypa0ZaaSaaaeaacaaIXaGaaGyoaaqaaiaaikdacaaI0aaa
% aiabgIKi7kaaicdacaGGUaGaaG4naiaaiMdacaGG7aaaaa!B488!
\[
m_y  = \int_{ - \infty }^\infty  {\int_{ - \infty }^\infty  {y \cdot f(x,y) \cdot dx \cdot dy} }  = \frac{1}{8}\left[ {\int\limits_0^2 {dx} \int\limits_0^x {ydy + } \int\limits_2^4 {dx} \int\limits_0^2 {ydy + \int\limits_4^6 {dx} \int\limits_0^1 {ydy} } } \right] = \frac{1}{8}\left[ {\frac{1}{2}\int\limits_0^2 {x^2 dx}  + \frac{1}{2}\int\limits_2^4 {4dx}  + \frac{1}{2}\int\limits_4^6 {1dx} } \right] = \frac{1}{8}\left[ {\frac{4}{3} + 4 + 1} \right] = \frac{{19}}{{24}} \approx 0.79;
\]
$

Дисперсии
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% qaaiaaicdaaeaacaWG4baaniabgUIiYdGcdaWdXbqaaiaadsgacaWG
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% WaaWbaaSqabeaacaaIYaaaaOGaamizaiaadMhacqGHRaWkdaWdXbqa
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% qCaeaacaWG5bWaaWbaaSqabeaacaaIYaaaaOGaamizaiaadMhaaSqa
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% beqaaiaaiodaaaGccaWGKbGaamiEaaWcbaGaaGimaaqaaiaaikdaa0
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% aeaacaaI4aGaamizaiaadIhaaSqaaiaaikdaaeaacaaI0aaaniabgU
% IiYdGccqGHRaWkdaWcaaqaaiaaigdaaeaacaaIZaaaamaapehabaGa
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% aSqabeaacaaIYaaaaOGaeyypa0ZaaSaaaeaacaaIXaaabaGaaGioaa
% aadaWadaqaamaalaaabaGaaGinaaqaaiaaiodaaaGaey4kaSYaaSaa
% aeaacaaIXaGaaGOnaaqaaiaaiodaaaGaey4kaSYaaSaaaeaacaaIYa
% aabaGaaG4maaaaaiaawUfacaGLDbaacqGHsislcaaIWaGaaiOlaiaa
% iAdacaaIYaGaeyypa0ZaaSaaaeaacaaIXaaabaGaaGioaaaacqGHfl
% Y1daWcaaqaaiaaikdacaaIYaaabaGaaG4maaaacqGHsislcaaIWaGa
% aiOlaiaaiAdacaaIYaGaeyisISRaaGimaiaac6cacaaIYaGaaGyoai
% aaiEdacaGG7aaaaaa!751F!
\[
\begin{array}{l}
 D_x  = \int_{ - \infty }^\infty  {\int_{ - \infty }^\infty  {x^2  \cdot f(x,y) \cdot dx \cdot dy} }  - m_x ^2  = \frac{1}{8}\left[ {\int\limits_0^2 {x^2 dx} \int\limits_0^x {dy + } \int\limits_2^4 {x^2 dx} \int\limits_0^2 {dy + \int\limits_4^6 {x^2 dx} \int\limits_0^1 {dy} } } \right] - m_x ^2  = \frac{1}{8}\left[ {\int\limits_0^2 {x^3 dx}  + 2\int\limits_2^4 {x^2 dx}  + \int\limits_4^6 {x^2 dx} } \right] - m_x ^2  = \frac{1}{8}\left[ {\frac{{16}}{4} + \frac{2}{3}\left( {64 - 8} \right) + \frac{1}{3}\left( {216 - 64} \right)} \right] - m_x ^2  =  \\ 
  = \frac{1}{8}\left( {\frac{{276}}{3}} \right) - 9.5 \approx 2.01; \\ 
 D_x  = \int_{ - \infty }^\infty  {\int_{ - \infty }^\infty  {y^2  \cdot f(x,y) \cdot dx \cdot dy} }  - m_y ^2  = \frac{1}{8}\left[ {\int\limits_0^2 {dx} \int\limits_0^x {y^2 dy + } \int\limits_2^4 {dx} \int\limits_0^2 {y^2 dy + \int\limits_4^6 {dx} \int\limits_0^1 {y^2 dy} } } \right] - m_y ^2  = \frac{1}{8}\left[ {\int\limits_0^2 {x^3 dx}  + \frac{1}{3}\int\limits_2^4 {8dx}  + \frac{1}{3}\int\limits_4^6 {dx} } \right] - m_x ^2  = \frac{1}{8}\left[ {\frac{4}{3} + \frac{{16}}{3} + \frac{2}{3}} \right] - 0.62 = \frac{1}{8} \cdot \frac{{22}}{3} - 0.62 \approx 0.297; \\ 
 \end{array}
\]
$


Среднеквадратические отклонения
$% MathType!MTEF!2!1!+-
% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9
% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x
% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS
% baaSqaaiaadIhaaeqaaOGaeyypa0ZaaOaaaeaacaWGebWaaSbaaSqa
% aiaadIhaaeqaaaqabaGccqGHijYUcaaIXaGaaiOlaiaaisdacaaIYa
% Gaai4oaaaa!414C!
\[
\sigma _x  = \sqrt {D_x }  \approx 1.42;
\]
$
$% MathType!MTEF!2!1!+-
% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
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% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9
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% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS
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% aiaadMhaaeqaaaqabaGccqGHijYUcaaIWaGaaiOlaiaaiwdacaaI0a
% Gaai4oaaaa!4150!
\[
\sigma _y  = \sqrt {D_y }  \approx 0.54;
\]
$

Корреляционный момент и коэффициент корреляции
$% MathType!MTEF!2!1!+-
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% aeaacaWG4bGaeyyXICTaamyEaiabgwSixlaadAgacaGGOaGaamiEai
% aacYcacaWG5bGaaiykaiabgwSixlaadsgacaWG4bGaeyyXICTaamiz
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% aiaaisdaaeaacaaIYaaaaiaadsgacaWG4baaleaacaaIYaaabaGaaG
% inaaqdcqGHRiI8aOGaey4kaSYaa8qCaeaadaWcaaqaaiaadIhaaeaa
% caaIYaaaaiaadsgacaWG4baaleaacaaI0aaabaGaaGOnaaqdcqGHRi
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% dacaGGUaGaaGimaiaaiwdacaaI1aGaai4oaaqaaaqaaiabeg8aYnaa
% BaaaleaacaWG4bGaamyEaaqabaGccqGH9aqpdaWcaaqaaiaadUeada
% WgaaWcbaGaamiEaiaadMhaaeqaaaGcbaGaeq4Wdm3aaSbaaSqaaiaa
% dMhaaeqaaOGaeq4Wdm3aaSbaaSqaaiaadIhaaeqaaaaakiabg2da9m
% aalaaabaGaeyOeI0IaaGimaiaac6cacaaIWaGaaGimaiaaiwdaaeaa
% caaIXaGaaiOlaiaaisdacaaIYaGaeyyXICTaaGimaiaac6cacaaI1a
% GaaGinaaaacqGH9aqpcqGHsislcaaIWaGaaiOlaiaaicdacaaI3aGa
% ai4oaaaaaa!F394!
\[
\begin{array}{l}
 K_{xy}  = \int_{ - \infty }^\infty  {\int_{ - \infty }^\infty  {x \cdot y \cdot f(x,y) \cdot dx \cdot dy} }  - \left( {m_x  \cdot m_x } \right) = \frac{1}{8}\left[ {\int\limits_0^2 {xdx} \int\limits_0^x {ydy + } \int\limits_2^4 {xdx} \int\limits_0^2 {ydy + \int\limits_4^6 {xdx} \int\limits_0^1 {ydy} } } \right] - \left( {m_x  \cdot m_x } \right) = \frac{1}{8}\left[ {\int\limits_0^2 {x\frac{{x^2 }}{2}dx}  + \int\limits_2^4 {x\frac{4}{2}dx}  + \int\limits_4^6 {\frac{x}{2}dx} } \right] - 2.43 = \frac{1}{8}\left[ {2 + 12 + 5} \right] - 2.43 =  - 0.055; \\ 
  \\ 
 \rho _{xy}  = \frac{{K_{xy} }}{{\sigma _y \sigma _x }} = \frac{{ - 0.005}}{{1.42 \cdot 0.54}} =  - 0.07; \\ 
 \end{array}
\]
$

 Профиль  
                  
 
 Re: ТерВер. Коэффициент корреляции
Сообщение11.12.2009, 01:37 
Заслуженный участник
Аватара пользователя


11/04/08
2748
Физтех
Вроде все верно, только мелкие опечатки есть в дисперсии для $D_y$ :)

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