Where Gaussian Random Distribution came from? I'm studying an introduction to probability, and before the explanation of the Central Limit Theorem the author presents the Gaussian Normal Distribution.
It has a complex and a non-intuitive formula, even if it's very important in the Probability field.
Was the gaussian distribution found before the deduction of the CLT ? How was this distribution discovered ? And what are the properties that makes it so special ?
 A: I want to comment on "It has a complex and a non-intuitive formula".
Let us first look at the reduced equation,
$$p(x)\propto e^{-x^2/2}.$$
This is simply the exponential of a square, nothing really difficult. The coefficient $2$ serves as a scaling factor to ensure that the standard deviation is $1$. (I won't give the technical details here as this involves improper integrals.)
Now we need to normalize the function so that the area under the curve is $1$ and the final formula is 
$$p(x)=\frac1{\sqrt{2\pi}}e^{-x^2/2}.$$
If we consider general mean $\mu$ and standard deviation $\sigma$, the formula is simply adapted by applying the linear transform $x\to(x-\mu)/\sigma$ that shifts $\mu$ to $0$ and rescales $\sigma$ to $1$. To ensure that the area remains $1$, the normalization factor needs to be multiplied by $\sigma$ to compensate. We have
$$p(x)=\frac1{\sqrt{2\pi}\sigma}e^{-(x-\mu)/2\sigma^2}.$$
If you look at the shape of the function, is very smooth, it is symmetrical, it has a single maximum and quickly decays on both sides. This behavior describes a random variable that remains "concentrated" around the mean.


To better understand the shape, you can look at the distributions of the sums of $n$ uniform random variables in the range $0,1$, for increasing $n$:

You are smoothing and smoothing the curve, which quickly converges to a Gaussian.
Now observe that with $x+y=z$, where $x,y$ follow a reduced Gaussian
$$\int_{-\infty}^\infty e^{-x^2/2}e^{-y^2/2}dx=\int_{-\infty}^\infty e^{-x^2/2}e^{-(z-x)^2/2}dx=\int_{-\infty}^\infty e^{-(x-z/2)^2/2-z^2/4}dx
\\=e^{-z^2/4}\int_{-\infty}^\infty e^{-x^2/2}dx\propto e^{-z^2/4}.$$
This proves that the distribution of the sum of two Gaussians is another Gaussian, with variance $2$. By a more complicated argument (based on the Fourier transform of convolutions), one can show the reciprocal: an additive distribution must be Gaussian.
A: Special and characteristic properties: 


*

*If $X$ has normal distribution then so has $aX+b$ where $a,b$ are constants with $a\neq0$. 

*If $X$ and $Y$ have joint normal distribution then so has $X+Y$.
Note that - if $X_n$ and $Y_n$ are sequences of random variables tending to Gaussians $X$ and $Y$ when $n\to\infty$ on base of CLT - also $X_n+Y_n$ will tend to a Gausian $Z$, so that $X+Y=Z$ where the LHS is a sum of Gaussians. This kind of explains why a summation of Gaussians is a Gaussian again, and tells us why Gaussian distribution is the only "candidate" for the CLT.
Similarly $aX_n$ will tend to a Gaussian, implying that $aX$ is Gaussian again.
A: Speaking to the historical question, it was "discovered" from an attempt to work out what distribution a sample mean should have, and there were some false starts on the way, but I'll leave it there because this isn't HSM.
The Gaussian distribution isn't that non-intuitive, if you look at it the right way. If the log-pdf has a Taylor series and we expand around the mode (which can be taken as $0$ after a translation), the lowest-order term that can survive is an $x^2/2!$ term. The integral $\int_\mathbb{R}\exp -x^2/2\, \text{d}x$ was obtained in the late 1700s (the proof everyone knows today took a little longer; the early arguments, though valid, weren't as short or clean), and it wasn't much harder to verify that the pdf proportional to $\exp -x^2/2$ is of variance $1$, which is a nice normalisation. The "complex" formula you're thinking of is just a linear generalisation, viz. $f(x)\mapsto \tfrac{1}{\sigma}f(\tfrac{x-\mu }{\sigma})$. This is of course slightly neater if you work with CDFs first, viz. $F(x)\mapsto F(\tfrac{x-\mu }{\sigma})$. 
There are several reasons Gaussian distributions can arise naturally, besides the CLT. These are discussed in my and others' answers here. My answer in particular discusses several problems to which a Gaussian is the unique or optimal solution.
