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### What is the purpose of $\frac{1}{\sigma \sqrt{2 \pi}}$ in $\frac{1}{\sigma \sqrt{2 \pi}}e^{\frac{(-(x - \mu ))^2}{2\sigma ^2}}$?

If you consider every possible outcome of some event you should expect the probability of it happening to be $1$, not $\sqrt{2\pi}$ so the constant scales the distribution to conform with the normal ...

### How was the normal distribution derived?

The Normal distribution came about from approximations of the binomial distribution (de Moivre), from linear regression (Gauss), and from the central limit theorem. The derivation given by Tim relates ...
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### Lower bound of Gaussian tail?

There is a standard lower estimate for $1-\Phi(x) = P(N(0,1)\ge x)$: for all $x>0$, $$1-\Phi(x)> \frac{x}{x^2+1}\varphi(x) = \frac{x}{x^2+1}\frac1{\sqrt{2\pi}}e^{-x^2/2}.$$ You can find the ...

### Gaussian distribution is isotropic?

I'd just like to add a bit of visuals to the other answers. When the variables are independent, i.e. the distrubtion is isotropic, it means that the distribution is aligned with the axis. For ...
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### Closed-form analytical solutions to Optimal Transport/Wasserstein distance

Although a bit old, this is indeed a good question. Here is my bit on the matter: Regarding Gaussian Mixture Models: A Wasserstein-type distance in the space of Gaussian Mixture Models, Julie Delon ...

I depends on the information you have and the quantities you want to get out. If you have the covariance matrices themselves then you should use the formula $$\Sigma_3 = \Sigma_1(\Sigma_1 + \Sigma_2)... 12 votes ### Uniform distribution on the surface of unit sphere Suppose X=(X_1, X_2, \ldots, X_n) iid and X_1 \sim N(0,1), then X \sim N(0, I_n), where N(0, I_n) is the multivariate normal distribution with zero-mean and identity covariance matrix. From ... 12 votes ### Proof of the affine property of normal distribution for a landscape matrix One should approach this through characteristic functions. Recall that X is normal N(\mu,\Sigma) if and only if, for every deterministic vector t of size N\times1,$$ E(\mathrm e^{\mathrm it'X}...
First of all as stated, the log is monotonically increasing so maximizing likelihood is equivalent to maximizing log likelihood. Furthermore, one can make use of $\ln(ab) = \ln(a) + \ln(b)$. Many ...