Понимаете тут в чем дело,
![$$\[{\mu _3} = \frac{1}{\lambda }\int\limits_{ - \infty }^{ + \infty } {{{\left( {x - m} \right)}^3}\exp \left( { - \frac{{x - \mu }}{\lambda } - \exp \left( { - \frac{{x - \mu }}{\lambda }} \right)} \right)dx} = \int\limits_0^{ + \infty } {{{\left( {\mu - \lambda \ln t - m} \right)}^3}\exp \left( { - t} \right)} dt\]$$](https://dxdy-04.korotkov.co.uk/f/3/6/0/360c77395c1ee53502139ac3487ecd2682.png "$$\[{\mu _3} = \frac{1}{\lambda }\int\limits_{ - \infty }^{ + \infty } {{{\left( {x - m} \right)}^3}\exp \left( { - \frac{{x - \mu }}{\lambda } - \exp \left( { - \frac{{x - \mu }}{\lambda }} \right)} \right)dx} = \int\limits_0^{ + \infty } {{{\left( {\mu - \lambda \ln t - m} \right)}^3}\exp \left( { - t} \right)} dt\]$$")
Замена:
![$\[t = \exp \left( { - \frac{{x - \mu }}{\lambda }} \right)\]$](https://dxdy-04.korotkov.co.uk/f/b/2/1/b217b2840b460432c6f302bc3f2e70e582.png "$\[t = \exp \left( { - \frac{{x - \mu }}{\lambda }} \right)\]$")
.
Интеграл справа расходится в нуле, а слева -- сходится. Я не знаю, в чем тут дело, но можно попробовать вычилить моменты через характеристическую функцию.
Характеристическая фукнция такой случайной величины:
![$\[\chi \left( t \right) = \Gamma \left( {1 - i\lambda t} \right){e^{i\mu t}}\]$](https://dxdy-01.korotkov.co.uk/f/c/a/1/ca1502dbb5cf900461765142c68e845282.png "$\[\chi \left( t \right) = \Gamma \left( {1 - i\lambda t} \right){e^{i\mu t}}\]$")
.
Третий центральный момент:
![$\[{\mu _3} = {m_3} - 3{m_2}{m_1} + 2m_1^3\]$](https://dxdy-01.korotkov.co.uk/f/8/7/a/87a7134ae2a06fdcb75d243d8211277582.png "$\[{\mu _3} = {m_3} - 3{m_2}{m_1} + 2m_1^3\]$")
.
По формуле
![$\[{m_n} = {\left( { - i} \right)^n}\frac{{{d^n}}}{{d{t^n}}}\chi \left( 0 \right)\]$](https://dxdy-04.korotkov.co.uk/f/f/2/2/f22e7e39db70a698743b7fc25cc77ffc82.png "$\[{m_n} = {\left( { - i} \right)^n}\frac{{{d^n}}}{{d{t^n}}}\chi \left( 0 \right)\]$")
можно попробовать вычислить три момента.
Например,
![$\[{m_1} = - i\frac{{d\chi }}{{dt}}\left( 0 \right) = - \lambda \Gamma '\left( 1 \right) + \mu = \lambda \gamma + \mu \]$](https://dxdy-04.korotkov.co.uk/f/3/a/6/3a61b6bd8ddf25619b5257f3b5152ee882.png "$\[{m_1} = - i\frac{{d\chi }}{{dt}}\left( 0 \right) = - \lambda \Gamma '\left( 1 \right) + \mu = \lambda \gamma + \mu \]$")
. Здесь
![$\[\gamma = - \Gamma '\left( 1 \right) \approx 0.57722\]$](https://dxdy-03.korotkov.co.uk/f/2/2/5/22563ecf4784eff1562f03980eba9efd82.png "$\[\gamma = - \Gamma '\left( 1 \right) \approx 0.57722\]$")
-- константа Эйлера.
Вот производные характеристической функции:
![$$\[\frac{{d\chi }}{{dt}} = - i\lambda \Gamma '\left( {1 - i\lambda t} \right){e^{i\mu t}} + i\mu \Gamma \left( {1 - i\lambda t} \right){e^{i\mu t}}\]$$](https://dxdy-02.korotkov.co.uk/f/d/4/a/d4a9212e6785dbc29688b41e3ffba3db82.png "$$\[\frac{{d\chi }}{{dt}} = - i\lambda \Gamma '\left( {1 - i\lambda t} \right){e^{i\mu t}} + i\mu \Gamma \left( {1 - i\lambda t} \right){e^{i\mu t}}\]$$")
![$$\[\frac{{{d^2}\chi }}{{d{t^2}}} = {\left( { - i\lambda } \right)^2}\Gamma ''\left( {1 - i\lambda t} \right){e^{i\mu t}} + 2\left( { - i\lambda } \right)\left( {i\mu } \right)\Gamma '\left( {1 - i\lambda t} \right){e^{i\mu t}} + {\left( {i\mu } \right)^2}\Gamma \left( {1 - i\lambda t} \right){e^{i\mu t}}\]$$](https://dxdy-03.korotkov.co.uk/f/a/0/8/a086a25a4040e262fbb2aed7c7b9fbc482.png "$$\[\frac{{{d^2}\chi }}{{d{t^2}}} = {\left( { - i\lambda } \right)^2}\Gamma ''\left( {1 - i\lambda t} \right){e^{i\mu t}} + 2\left( { - i\lambda } \right)\left( {i\mu } \right)\Gamma '\left( {1 - i\lambda t} \right){e^{i\mu t}} + {\left( {i\mu } \right)^2}\Gamma \left( {1 - i\lambda t} \right){e^{i\mu t}}\]$$")
![$$\[\begin{gathered}
\frac{{{d^3}\chi }}{{d{t^3}}} = {\left( { - i\lambda } \right)^3}\Gamma '''\left( {1 - i\lambda t} \right){e^{i\mu t}} + 3{\left( { - i\lambda } \right)^2}\left( {i\mu } \right)\Gamma ''\left( {1 - i\lambda t} \right){e^{i\mu t}} + \hfill \\
{\text{ }} + 3\left( { - i\lambda } \right){\left( {i\mu } \right)^2}\Gamma '\left( {1 - i\lambda t} \right){e^{i\mu t}} + {\left( {i\mu } \right)^3}\Gamma \left( {1 - i\lambda t} \right){e^{i\mu t}} \hfill \\
\end{gathered} \]$$](https://dxdy-03.korotkov.co.uk/f/e/c/3/ec3a63ae5ef919d94b1f118f209926c082.png "$$\[\begin{gathered}
\frac{{{d^3}\chi }}{{d{t^3}}} = {\left( { - i\lambda } \right)^3}\Gamma '''\left( {1 - i\lambda t} \right){e^{i\mu t}} + 3{\left( { - i\lambda } \right)^2}\left( {i\mu } \right)\Gamma ''\left( {1 - i\lambda t} \right){e^{i\mu t}} + \hfill \\
{\text{ }} + 3\left( { - i\lambda } \right){\left( {i\mu } \right)^2}\Gamma '\left( {1 - i\lambda t} \right){e^{i\mu t}} + {\left( {i\mu } \right)^3}\Gamma \left( {1 - i\lambda t} \right){e^{i\mu t}} \hfill \\
\end{gathered} \]$$")
Подставьте везде
. Вот значения производных гамма-функций в единице:
![$\[\begin{gathered}
\Gamma '\left( 1 \right) = - \gamma \hfill \\
\Gamma ''\left( 1 \right) = {\gamma ^2} + \frac{{{\pi ^2}}}{6} \hfill \\
\Gamma '''\left( 1 \right) = - 2\zeta \left( 3 \right) - {\gamma ^3} - \gamma \frac{{{\pi ^2}}}{2} \hfill \\
\end{gathered} \]$](https://dxdy-04.korotkov.co.uk/f/3/8/d/38db77dc089268d8bc29966b1f2c713482.png "$\[\begin{gathered}
\Gamma '\left( 1 \right) = - \gamma \hfill \\
\Gamma ''\left( 1 \right) = {\gamma ^2} + \frac{{{\pi ^2}}}{6} \hfill \\
\Gamma '''\left( 1 \right) = - 2\zeta \left( 3 \right) - {\gamma ^3} - \gamma \frac{{{\pi ^2}}}{2} \hfill \\
\end{gathered} \]$")
Свойства гамма-функции можете посмотреть, например, здесь:
http://functions.wolfram.com/GammaBetaErf/Gamma/Попробуйте хоть так.
21.07.2011, 16:03 
. 
, 
,
!
![$$\[\begin{gathered}
\int\limits_0^{ + \infty } {{{\left( {\mu - m - \lambda \ln t} \right)}^3}\exp \left( { - t} \right)dt} = {\left( {\mu - m} \right)^3} - 3{\left( {\mu - m} \right)^2}\lambda \int\limits_0^{ + \infty } {\ln t\exp \left( { - t} \right)dt} + \hfill \\
{\text{ }} + 3\left( {\mu - m} \right){\lambda ^2}\int\limits_0^{ + \infty } {{{\ln }^2}t\exp \left( { - t} \right)dt} - {\lambda ^3}\int\limits_0^{ + \infty } {{{\ln }^3}t\exp \left( { - t} \right)dt} \hfill \\
\end{gathered} \]$$](https://dxdy-04.korotkov.co.uk/f/3/6/1/361b388903c1a4413f57352b57c2834c82.png "$$\[\begin{gathered}
\int\limits_0^{ + \infty } {{{\left( {\mu - m - \lambda \ln t} \right)}^3}\exp \left( { - t} \right)dt} = {\left( {\mu - m} \right)^3} - 3{\left( {\mu - m} \right)^2}\lambda \int\limits_0^{ + \infty } {\ln t\exp \left( { - t} \right)dt} + \hfill \\
{\text{ }} + 3\left( {\mu - m} \right){\lambda ^2}\int\limits_0^{ + \infty } {{{\ln }^2}t\exp \left( { - t} \right)dt} - {\lambda ^3}\int\limits_0^{ + \infty } {{{\ln }^3}t\exp \left( { - t} \right)dt} \hfill \\
\end{gathered} \]$$")
![$
\[\begin{gathered}
\int\limits_0^{ + \infty } {\ln t\exp \left( { - t} \right)dt} = - \gamma \hfill \\
\int\limits_0^{ + \infty } {{{\ln }^2}t\exp \left( { - t} \right)dt} = {\gamma ^2} + \frac{{{\pi ^2}}}{6} \hfill \\
\int\limits_0^{ + \infty } {{{\ln }^3}t\exp \left( { - t} \right)dt} = - 2\zeta \left( 3 \right) - {\gamma ^3} - \gamma \frac{{{\pi ^2}}}{2} \hfill \\
\end{gathered} \]$](https://dxdy-04.korotkov.co.uk/f/f/f/4/ff4de41b09536cad354be19d1a7bb3b182.png "$
\[\begin{gathered}
\int\limits_0^{ + \infty } {\ln t\exp \left( { - t} \right)dt} = - \gamma \hfill \\
\int\limits_0^{ + \infty } {{{\ln }^2}t\exp \left( { - t} \right)dt} = {\gamma ^2} + \frac{{{\pi ^2}}}{6} \hfill \\
\int\limits_0^{ + \infty } {{{\ln }^3}t\exp \left( { - t} \right)dt} = - 2\zeta \left( 3 \right) - {\gamma ^3} - \gamma \frac{{{\pi ^2}}}{2} \hfill \\
\end{gathered} \]$")
![$\[m = \mu + \lambda \gamma \]$](https://dxdy-03.korotkov.co.uk/f/e/0/c/e0cb0ce750d7352dd3d70b1e74b3fdd282.png "$\[m = \mu + \lambda \gamma \]$")
, тривиально получается ответ.
" при записи параметра экспоненциальной функции. 
