axjack's blog

axjack is said to be an abbreviation for An eXistent JApanese Cool Klutz.

コーシー分布

準備:微分の公式など

\displaystyle (\frac{f}{g})' = \frac{f'g - fg'}{g^2} \\
\displaystyle (\sin \theta)' = \cos \theta \\ 
\displaystyle (\cos \theta)' = -\sin \theta \\
\displaystyle (\tan \theta)' = (\frac{\sin \theta}{\cos \theta} )' = \frac{(\sin \theta)'\cos \theta - \sin \theta(\cos \theta)'}{\cos^{2} \theta}  = \frac{ 1 }{ \cos^{2} \theta } \\
\displaystyle ( 1 + \tan^{2} \theta)^{-1}  = ( \frac{1}{\cos^2 \theta} )^{-1} = \cos^{2} \theta

コーシー分布の積分

 \displaystyle f(x) = \frac{1}{ \pi ( 1 + x^2)  } x \in \mathbb{R} 積分すると、 \displaystyle x = \tan \thetaと置換し、

\displaystyle  \int^{\infty}_{-\infty} \frac{ 1 }{ \pi( 1 + x^2 ) } dx  =   \int^{\frac{\pi}{2}}_{-\frac{\pi}{2}}  \frac{1}{\pi} \cos^{2} \theta \frac{1}{\cos^2 \theta } d\theta  = 
  \int^{\frac{\pi}{2}}_{-\frac{\pi}{2}}  \frac{1}{\pi} d\theta = \frac{1}{\pi}  \left[   \theta \right]^{\frac{\pi}{2}}_{-\frac{\pi}{2}}  = \frac{1}{\pi} ( \frac{\pi }{ 2} - \frac{-\pi}{2})  = 1


となる。

axjack is said to be an abbreviation for An eXistent JApanese Cool Klutz.