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Wednesday, June 04, 2014

 

Why area under a normal distribution sums to 1 ?

Have you ever wondered why
  12πex22dx=1
? That is, why
  ex22dx=2π
? Let
  I=ex22dxII=(ex22dx)(ey22dy)I2=e12(x2+y2)dxdy
The double integral is easier to resolve in polar coordinate. In general,
 
In this particular case,
  \begin{align*} g(x,y) &= e^{-\frac{1}{2}(x^2+y^2)} \\ \mathbf{T}(r,\theta) &= (r\cos{\theta}, r\sin{\theta}) = (x, y) \\ g(\mathbf{T}(r,\theta)) &= e^{-\frac{1}{2}((r\cos{\theta})^2+(r\sin{\theta})^2)} = e^{-\frac{1}{2}r^2} \\ D \mathbf{T}(r,\theta) &= \begin{bmatrix}\frac{d}{dr}r\cos{\theta} & \frac{d}{d\theta}r\cos{\theta}\\\frac{d}{dr}r\sin{\theta} & \frac{d}{d\theta}r\sin{\theta}\end{bmatrix} = \begin{bmatrix}\cos{\theta} & -r\sin{\theta}\\\sin{\theta} & r\cos{\theta}\end{bmatrix} \\ \big| \det D \mathbf{T}(r,\theta)\,\big| &= \begin{vmatrix}\cos{\theta} & -r\sin{\theta}\\\sin{\theta} & r\cos{\theta}\end{vmatrix} = r \\ \end{align*}
Therefore
  \begin{aligned} \mathbf{I}^2 &= \int_0^{2\pi} \int_0^{\infty} e^{-\frac{1}{2}r^2} r\,dr\,d\theta \\ \end{aligned}
Let u = \frac{1}{2}r^2, so du = r\,dr
  \begin{aligned} \mathbf{I}^2 &= \int_0^{2\pi} \int_0^{\infty} e^{-u} \,du\,d\theta = \int_0^{2\pi} -e^{-u} \bigg|_{u=0}^{\infty} \,d\theta = \int_0^{2\pi}d\theta = 2\pi \\ \therefore \color{blue}{\mathbf{I}} &\color{blue}{= \sqrt{2\pi}} \\ \end{aligned}
\Box

References:


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