Ok day. Chem, Bio, ML

 Added 2022-07-30

3h Studying chem and bio, 2h Gaussian integrals (learned several cool tricks!), drafted a blog post.

What I did

Gaussian integral trick

Regarding Gaussian integrals, a cool way to derive the moments of a normal distribution is by applying the differential operator 2KddK2K \frac{d}{dK}

(2K2ddK)m[e−z22K]=(2K2)(z22K2e−z22K)=z2e−z22K\begin{aligned} \left(2K^2 \frac{d}{dK}\right)^m\left[e^{-\frac{z^2}{2K}}\right] &= (2K^2)\left(\frac{z^2}{2K^2} e^{-\frac{z^2}{2K}}\right) \\ &= z^2 e^{-\frac{z^2}{2K}} \end{aligned}

Because z2z^2 is constant applying the operator mm times brings down a z2z^2 each time, giving z2mz^{2m} as desired for the expectation E[z2m]E[z^{2m}]

E[z2m]=12πK∫−∞∞z2me−z22Kdz=12πK(2K2ddK)m∫−∞∞e−z22Kdz=12πK(2K2ddK)m[2πK]\begin{aligned} E[z^{2m}] &= \frac{1}{\sqrt{2\pi K}} \int_{-\infty}^{\infty} z^{2m} e^{-\frac{z^2}{2K}}dz \\ &= \frac{1}{\sqrt{2\pi K}} \left(2K^2 \frac{d}{dK}\right)^m \int_{-\infty}^{\infty} e^{-\frac{z^2}{2K}}dz \\ &= \frac{1}{\sqrt{2\pi K}} \left(2K^2 \frac{d}{dK}\right)^m\left[\sqrt{2\pi K}\right] \end{aligned}

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