ТерВер. Коэффициент корреляции

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Проверьте, пожалуйста, решение задачи.

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


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! \[ \rho _{xy} = \frac{{K_{xy} }}{{\sigma _y \sigma _x }} = \frac{{ - 2.005}}{{1.58 \cdot 0.95}} = - 1.33; \] $

[ИСН]

Ничего не читал, кроме последней строчки, но такие коэффициенты корреляции бывают только в военное время.

[ShMaxG]

Ошибки в обоих матожиданиях.

[new_sergei]

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.

[PAV]

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:

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

[--mS--]

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: $\int_0^2\,dy \int_y^4\,dx + \int_0^1\,dy \int_4^6\,dx$

[new_sergei]

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!+- % feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaaBa % aaleaacaWG5baabeaakiabg2da9maapedabaWaa8qmaeaacaWG5bGa % eyyXICTaamOzaiaacIcacaWG4bGaaiilaiaadMhacaGGPaGaeyyXIC % TaamizaiaadIhacqGHflY1caWGKbGaamyEaaWcbaGaeyOeI0IaeyOh % IukabaGaeyOhIukaniabgUIiYdaaleaacqGHsislcqGHEisPaeaacq % GHEisPa0Gaey4kIipakiabg2da9maalaaabaGaaGymaaqaaiaaiIda % aaWaamWaaeaadaWdXbqaaiaadsgacaWG4baaleaacaaIWaaabaGaaG % OmaaqdcqGHRiI8aOWaa8qCaeaacaWG5bGaamizaiaadMhacqGHRaWk % aSqaaiaaicdaaeaacaWG4baaniabgUIiYdGcdaWdXbqaaiaadsgaca % WG4baaleaacaaIYaaabaGaaGinaaqdcqGHRiI8aOWaa8qCaeaacaWG % 5bGaamizaiaadMhacqGHRaWkdaWdXbqaaiaadsgacaWG4baaleaaca % aI0aaabaGaaGOnaaqdcqGHRiI8aOWaa8qCaeaacaWG5bGaamizaiaa % dMhaaSqaaiaaicdaaeaacaaIXaaaniabgUIiYdaaleaacaaIWaaaba % GaaGOmaaqdcqGHRiI8aaGccaGLBbGaayzxaaGaeyypa0ZaaSaaaeaa % caaIXaaabaGaaGioaaaadaWadaqaamaalaaabaGaaGymaaqaaiaaik % daaaWaa8qCaeaacaWG4bWaaWbaaSqabeaacaaIYaaaaOGaamizaiaa % dIhaaSqaaiaaicdaaeaacaaIYaaaniabgUIiYdGccqGHRaWkdaWcaa % qaaiaaigdaaeaacaaIYaaaamaapehabaGaaGinaiaadsgacaWG4baa % leaacaaIYaaabaGaaGinaaqdcqGHRiI8aOGaey4kaSYaaSaaaeaaca % 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
$% MathType!MTEF!2!1!+- % feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGeb % WaaSbaaSqaaiaadIhaaeqaaOGaeyypa0Zaa8qmaeaadaWdXaqaaiaa % dIhadaahaaWcbeqaaiaaikdaaaGccqGHflY1caWGMbGaaiikaiaadI % hacaGGSaGaamyEaiaacMcacqGHflY1caWGKbGaamiEaiabgwSixlaa % dsgacaWG5baaleaacqGHsislcqGHEisPaeaacqGHEisPa0Gaey4kIi % paaSqaaiabgkHiTiabg6HiLcqaaiabg6HiLcqdcqGHRiI8aOGaeyOe % I0IaamyBamaaBaaaleaacaWG4baabeaakmaaCaaaleqabaGaaGOmaa % aakiabg2da9maalaaabaGaaGymaaqaaiaaiIdaaaWaamWaaeaadaWd % XbqaaiaadIhadaahaaWcbeqaaiaaikdaaaGccaWGKbGaamiEaaWcba % GaaGimaaqaaiaaikdaa0Gaey4kIipakmaapehabaGaamizaiaadMha % cqGHRaWkaSqaaiaaicdaaeaacaWG4baaniabgUIiYdGcdaWdXbqaai % aadIhadaahaaWcbeqaaiaaikdaaaGccaWGKbGaamiEaaWcbaGaaGOm % aaqaaiaaisdaa0Gaey4kIipakmaapehabaGaamizaiaadMhacqGHRa % WkdaWdXbqaaiaadIhadaahaaWcbeqaaiaaikdaaaGccaWGKbGaamiE % aaWcbaGaaGinaaqaaiaaiAdaa0Gaey4kIipakmaapehabaGaamizai % aadMhaaSqaaiaaicdaaeaacaaIXaaaniabgUIiYdaaleaacaaIWaaa % baGaaGOmaaqdcqGHRiI8aaGccaGLBbGaayzxaaGaeyOeI0IaamyBam % aaBaaaleaacaWG4baabeaakmaaCaaaleqabaGaaGOmaaaakiabg2da % 9maalaaabaGaaGymaaqaaiaaiIdaaaWaamWaaeaadaWdXbqaaiaadI % hadaahaaWcbeqaaiaaiodaaaGccaWGKbGaamiEaaWcbaGaaGimaaqa % aiaaikdaa0Gaey4kIipakiabgUcaRiaaikdadaWdXbqaaiaadIhada % ahaaWcbeqaaiaaikdaaaGccaWGKbGaamiEaaWcbaGaaGOmaaqaaiaa % isdaa0Gaey4kIipakiabgUcaRmaapehabaGaamiEamaaCaaaleqaba % GaaGOmaaaakiaadsgacaWG4baaleaacaaI0aaabaGaaGOnaaqdcqGH % RiI8aaGccaGLBbGaayzxaaGaeyOeI0IaamyBamaaBaaaleaacaWG4b % aabeaakmaaCaaaleqabaGaaGOmaaaakiabg2da9maalaaabaGaaGym % aaqaaiaaiIdaaaWaamWaaeaadaWcaaqaaiaaigdacaaI2aaabaGaaG % inaaaacqGHRaWkdaWcaaqaaiaaikdaaeaacaaIZaaaamaabmaabaGa % aGOnaiaaisdacqGHsislcaaI4aaacaGLOaGaayzkaaGaey4kaSYaaS % aaaeaacaaIXaaabaGaaG4maaaadaqadaqaaiaaikdacaaIXaGaaGOn % aiabgkHiTiaaiAdacaaI0aaacaGLOaGaayzkaaaacaGLBbGaayzxaa % GaeyOeI0IaamyBamaaBaaaleaacaWG4baabeaakmaaCaaaleqabaGa % aGOmaaaakiabg2da9aqaaiabg2da9maalaaabaGaaGymaaqaaiaaiI % daaaWaaeWaaeaadaWcaaqaaiaaikdacaaI3aGaaGOnaaqaaiaaioda % aaaacaGLOaGaayzkaaGaeyOeI0IaaGyoaiaac6cacaaI1aGaeyisIS % RaaGOmaiaac6cacaaIWaGaaGymaiaacUdaaeaacaWGebWaaSbaaSqa % aiaadIhaaeqaaOGaeyypa0Zaa8qmaeaadaWdXaqaaiaadMhadaahaa % WcbeqaaiaaikdaaaGccqGHflY1caWGMbGaaiikaiaadIhacaGGSaGa % amyEaiaacMcacqGHflY1caWGKbGaamiEaiabgwSixlaadsgacaWG5b % aaleaacqGHsislcqGHEisPaeaacqGHEisPa0Gaey4kIipaaSqaaiab % gkHiTiabg6HiLcqaaiabg6HiLcqdcqGHRiI8aOGaeyOeI0IaamyBam % aaBaaaleaacaWG5baabeaakmaaCaaaleqabaGaaGOmaaaakiabg2da % 9maalaaabaGaaGymaaqaaiaaiIdaaaWaamWaaeaadaWdXbqaaiaads % gacaWG4baaleaacaaIWaaabaGaaGOmaaqdcqGHRiI8aOWaa8qCaeaa % caWG5bWaaWbaaSqabeaacaaIYaaaaOGaamizaiaadMhacqGHRaWkaS % qaaiaaicdaaeaacaWG4baaniabgUIiYdGcdaWdXbqaaiaadsgacaWG % 4baaleaacaaIYaaabaGaaGinaaqdcqGHRiI8aOWaa8qCaeaacaWG5b % WaaWbaaSqabeaacaaIYaaaaOGaamizaiaadMhacqGHRaWkdaWdXbqa % aiaadsgacaWG4baaleaacaaI0aaabaGaaGOnaaqdcqGHRiI8aOWaa8 % qCaeaacaWG5bWaaWbaaSqabeaacaaIYaaaaOGaamizaiaadMhaaSqa % aiaaicdaaeaacaaIXaaaniabgUIiYdaaleaacaaIWaaabaGaaGOmaa % qdcqGHRiI8aaGccaGLBbGaayzxaaGaeyOeI0IaamyBamaaBaaaleaa % caWG5baabeaakmaaCaaaleqabaGaaGOmaaaakiabg2da9maalaaaba % GaaGymaaqaaiaaiIdaaaWaamWaaeaadaWdXbqaaiaadIhadaahaaWc % beqaaiaaiodaaaGccaWGKbGaamiEaaWcbaGaaGimaaqaaiaaikdaa0 % Gaey4kIipakiabgUcaRmaalaaabaGaaGymaaqaaiaaiodaaaWaa8qC % aeaacaaI4aGaamizaiaadIhaaSqaaiaaikdaaeaacaaI0aaaniabgU % IiYdGccqGHRaWkdaWcaaqaaiaaigdaaeaacaaIZaaaamaapehabaGa % amizaiaadIhaaSqaaiaaisdaaeaacaaI2aaaniabgUIiYdaakiaawU % facaGLDbaacqGHsislcaWGTbWaaSbaaSqaaiaadIhaaeqaaOWaaWba % 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 % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS % baaSqaaiaadMhaaeqaaOGaeyypa0ZaaOaaaeaacaWGebWaaSbaaSqa % aiaadMhaaeqaaaqabaGccqGHijYUcaaIWaGaaiOlaiaaiwdacaaI0a % Gaai4oaaaa!4150! \[ \sigma _y = \sqrt {D_y } \approx 0.54; \] $

Корреляционный момент и коэффициент корреляции
$% MathType!MTEF!2!1!+- % feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGlb % WaaSbaaSqaaiaadIhacaWG5baabeaakiabg2da9maapedabaWaa8qm % aeaacaWG4bGaeyyXICTaamyEaiabgwSixlaadAgacaGGOaGaamiEai % aacYcacaWG5bGaaiykaiabgwSixlaadsgacaWG4bGaeyyXICTaamiz % aiaadMhaaSqaaiabgkHiTiabg6HiLcqaaiabg6HiLcqdcqGHRiI8aa % WcbaGaeyOeI0IaeyOhIukabaGaeyOhIukaniabgUIiYdGccqGHsisl % daqadaqaaiaad2gadaWgaaWcbaGaamiEaaqabaGccqGHflY1caWGTb % WaaSbaaSqaaiaadIhaaeqaaaGccaGLOaGaayzkaaGaeyypa0ZaaSaa % aeaacaaIXaaabaGaaGioaaaadaWadaqaamaapehabaGaamiEaiaads % gacaWG4baaleaacaaIWaaabaGaaGOmaaqdcqGHRiI8aOWaa8qCaeaa % caWG5bGaamizaiaadMhacqGHRaWkaSqaaiaaicdaaeaacaWG4baani % abgUIiYdGcdaWdXbqaaiaadIhacaWGKbGaamiEaaWcbaGaaGOmaaqa % aiaaisdaa0Gaey4kIipakmaapehabaGaamyEaiaadsgacaWG5bGaey % 4kaSYaa8qCaeaacaWG4bGaamizaiaadIhaaSqaaiaaisdaaeaacaaI % 2aaaniabgUIiYdGcdaWdXbqaaiaadMhacaWGKbGaamyEaaWcbaGaaG % imaaqaaiaaigdaa0Gaey4kIipaaSqaaiaaicdaaeaacaaIYaaaniab % gUIiYdaakiaawUfacaGLDbaacqGHsisldaqadaqaaiaad2gadaWgaa % WcbaGaamiEaaqabaGccqGHflY1caWGTbWaaSbaaSqaaiaadIhaaeqa % aaGccaGLOaGaayzkaaGaeyypa0ZaaSaaaeaacaaIXaaabaGaaGioaa % aadaWadaqaamaapehabaGaamiEamaalaaabaGaamiEamaaCaaaleqa % baGaaGOmaaaaaOqaaiaaikdaaaGaamizaiaadIhaaSqaaiaaicdaae % aacaaIYaaaniabgUIiYdGccqGHRaWkdaWdXbqaaiaadIhadaWcaaqa % aiaaisdaaeaacaaIYaaaaiaadsgacaWG4baaleaacaaIYaaabaGaaG % inaaqdcqGHRiI8aOGaey4kaSYaa8qCaeaadaWcaaqaaiaadIhaaeaa % caaIYaaaaiaadsgacaWG4baaleaacaaI0aaabaGaaGOnaaqdcqGHRi % I8aaGccaGLBbGaayzxaaGaeyOeI0IaaGOmaiaac6cacaaI0aGaaG4m % aiabg2da9maalaaabaGaaGymaaqaaiaaiIdaaaWaamWaaeaacaaIYa % Gaey4kaSIaaGymaiaaikdacqGHRaWkcaaI1aaacaGLBbGaayzxaaGa % eyOeI0IaaGOmaiaac6cacaaI0aGaaG4maiabg2da9iabgkHiTiaaic % dacaGGUaGaaGimaiaaiwdacaaI1aGaai4oaaqaaaqaaiabeg8aYnaa % BaaaleaacaWG4bGaamyEaaqabaGccqGH9aqpdaWcaaqaaiaadUeada % WgaaWcbaGaamiEaiaadMhaaeqaaaGcbaGaeq4Wdm3aaSbaaSqaaiaa % dMhaaeqaaOGaeq4Wdm3aaSbaaSqaaiaadIhaaeqaaaaakiabg2da9m % aalaaabaGaeyOeI0IaaGimaiaac6cacaaIWaGaaGimaiaaiwdaaeaa % caaIXaGaaiOlaiaaisdacaaIYaGaeyyXICTaaGimaiaac6cacaaI1a % GaaGinaaaacqGH9aqpcqGHsislcaaIWaGaaiOlaiaaicdacaaI3aGa % ai4oaaaaaa!F394! \[ \begin{array}{l} K_{xy} = \int_{ - \infty }^\infty {\int_{ - \infty }^\infty {x \cdot y \cdot f(x,y) \cdot dx \cdot dy} } - \left( {m_x \cdot m_x } \right) = \frac{1}{8}\left[ {\int\limits_0^2 {xdx} \int\limits_0^x {ydy + } \int\limits_2^4 {xdx} \int\limits_0^2 {ydy + \int\limits_4^6 {xdx} \int\limits_0^1 {ydy} } } \right] - \left( {m_x \cdot m_x } \right) = \frac{1}{8}\left[ {\int\limits_0^2 {x\frac{{x^2 }}{2}dx} + \int\limits_2^4 {x\frac{4}{2}dx} + \int\limits_4^6 {\frac{x}{2}dx} } \right] - 2.43 = \frac{1}{8}\left[ {2 + 12 + 5} \right] - 2.43 = - 0.055; \\ \\ \rho _{xy} = \frac{{K_{xy} }}{{\sigma _y \sigma _x }} = \frac{{ - 0.005}}{{1.42 \cdot 0.54}} = - 0.07; \\ \end{array} \] $

[ShMaxG]

Вроде все верно, только мелкие опечатки есть в дисперсии для $D_y$