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I have a very simple question regarding normal distribution, which nonetheless bugs me a bit. In my reference book related to Bayesian data analysis the author seems to use the following results of the book:

$$\int_{-\infty}^{\infty} x \,g(x)\,dx = c\mu\sqrt{2\pi\sigma^2}\;\;\;\;\;\;(1)$$ $$\int_{-\infty}^{\infty} (x-\mu)^2 \,g(x)\,dx = c\sigma^2\sqrt{2\pi\sigma^2}\;\;\;\;\;\;(2)$$

$$\int_{-\infty}^{\infty} \,g(x)\,dx = c\sqrt{2\pi\sigma^2}\;\;\;\;\;\;(3)$$

where $c, \mu, \sigma^2$ are constant values to write $g(x)$ as the Gaussian distribution:

$$p(x)=\frac{1}{\sqrt{2\pi\sigma^2}}\exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right).$$

I'm not fully certain how the author got from $(1)$, $(2)$ and $(3)$ to $p(x)$, but I am guessing proportionality plays a part here? Do you think the author defines (without explicitly stating it) that:

$$p(x) = \frac{g(x)}{c\sqrt{2\pi\sigma^2}}?$$

To answer user @Stockfish the function $g(x)$ is actually defined as:

$$g(x) = \int\exp\left(-\frac{\beta}{2}\{x-\mu-\textbf{g}^T\textbf{v}\}^2-\frac12\textbf{v}^T\textbf{A}\textbf{v}\right)\,d\textbf{v},$$

where $\beta$ is a constant, $\textbf{g}, \textbf{v}\in R^W$ and $\textbf{A}$ is a symmetric $W\times W$ real matrix.

UPDATE:

This post is actually a follow up question to my previous post (if it helps):

How to evaluate the integral $\int\mathbf{g}^T\mathbf{v} \exp{(-\frac{1}{2}\mathbf{v}^T \mathbf{A}\mathbf{v})}\,d\mathbf{v}$

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  • $\begingroup$ Do you mean $g(x) = e^{-(x-\mu)^2}$? $\endgroup$ – Stockfish Oct 24 '18 at 13:03
  • $\begingroup$ @LeeDavidChungLin thank you for that, sorry, my mistake. $\endgroup$ – jjepsuomi Oct 24 '18 at 13:03
  • $\begingroup$ @Stockfish I edited to post. Sorry for any confusion in my question, I'm a bit confused myself also because the author drops out many constants so I have to guess what he is doing in the derivation. $\endgroup$ – jjepsuomi Oct 24 '18 at 13:11
  • $\begingroup$ Urgh :D but if $p(x) = g(x)/...$ we had $\int x g(x) dx = 0$, so that cannot be. $\endgroup$ – Stockfish Oct 24 '18 at 13:18
  • $\begingroup$ @Stockfish thank you for your help :) Now I'm even more confused xD $\endgroup$ – jjepsuomi Oct 24 '18 at 13:22
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Do you think the author defines (without explicitly stating it) that: $$p(x) = \frac{g(x)}{c\sqrt{2\pi\sigma^2}}$$

Yes, you are right, as you can see from the first three integrals, $$\int_{-\infty}^{\infty} x \,g(x)\,dx = k E(X)$$ $$\int_{-\infty}^{\infty} (x-\mu)^2 \,g(x)\,dx = k \text{var}(X) $$ $$\int_{-\infty}^{\infty} \,g(x)\,dx = k(1)$$ where $k = c\sqrt{2\pi\sigma^2}$. That being said, it is now easy to see why $g(x) = k p(x)$, where $p(x)$ is the PDF of $\mathcal{N}(\mu,\sigma^2)$

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    $\begingroup$ Excellent! Thank you very much =) $\endgroup$ – jjepsuomi Oct 24 '18 at 14:07
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I think I got it myself also, it seems to be as I suspected. If you evaluate the integral I gave above ($g(x)$ that is), you will get:

$$g(x) = (2\pi)^{W/2}|(\textbf{A}+\beta\textbf{g}\textbf{g}^T)|^{-1/2} \exp\left(-\frac{1}{2\sigma^2} (x-\mu)^2 \right),$$

where $\sigma^2 = \frac{1}{\beta-\beta^2\textbf{g}^T(\textbf{A}+\beta\textbf{g}\textbf{g}^T)^{-1}\textbf{g}}=\frac1\beta+\textbf{g}^T\textbf{A}^{-1}\textbf{g}$ and $c=(2\pi)^{W/2}|(\textbf{A}+\beta\textbf{g}\textbf{g}^T)|^{-1/2}.$

So we have $p(x)$ as:

$$p(x) = \frac{g(x)}{c\sqrt{2\pi\sigma^2}}=\frac{1}{\sqrt{2\pi\sigma^2}}\exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right).$$

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